LB 


UC-NRI 


An  Investigation  of  Certain  Abilities 

Fundamental  to  the  Study  of 

Geometry 


BY 


JOHN  HARRISON  MINNICK 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL  IN 

PARTIAL  FULFILLMENT  OF  THE  REQUIREMENTS  FOR  THE 

DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


PRESS  OF 

THE  NEW  ERA  PRINTING  CCMPA-NY 
LANCASTER,  PA. 


Ipl8 


EXCHANGE 


i 


UNIVERSITY  OF  PENNSYLVANIA 


An  Investigation  of  Certain  Abilities 

Fundamental  to  the  Study  of 

Geometry 


BY 

JOHN  HARRISON  MINNICK 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL  IN 

PARTIAL  FULFILLMENT  OF  THE  REQUIREMENTS  FOR  THE 

DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER,  PA. 


1918 


ACKNOWLEDGMENTS 


I  wish  to  acknowledge  my  indebtedness  to  those  whose  aid 
has  made  this  study  possible.  Dean  H.  L.  Smith  of  Indiana 
University,  Mr.  E.  E.  Arnold  of  Albany,  New  York,  and  many 
superintendents,  principals  and  teachers  throughout  the  country 
gave  valuable  assistance  in  gathering  data.  Doctors  Ralph  and 
Robert  Duncan  of  the  University  of  Pennsylvania,  Mr.  Roy 
Cumins  of  the  Philadelphia  Public  Schools  and  Miss  Jessie  A. 
Smith  of  Indiana  University  read  the  manuscript  and  offered 
many  useful  suggestions.  Especially  am  I  indebted  to  Pro- 
fessor A.  Duncan  Yocum  for  valuable  assistance  given  through- 
out the  entire  investigation.  I  wish  also  to  acknowledge  my 
indebtedness  to  my  wife,  without  whose  help  with  the  many 
details  of  the  work  this  study  would  have  been  impossible. 

J.  H.  M. 


381678 


TABLE  OF  CONTENTS 


PAGE 

Acknowledgments iii 

Table  of  Contents v 

Introduction vii 

Part  I. 

Purpose  of  the  Investigation I 

Plan  of  Procedure I 

Giving  the  Tests 2 

Scoring  the  Papers 2 

Criticism  of  the  Tests 3 

Criticism  of  Teachers'  Grades 4 

Coefficients  of  Correlation 4 

Standards  of  Achievements 4 

Part  II. 

I.  Purpose  of  the  Investigation 6 

II.  Brief  Statement  of  the  Plan  of  Procedure 7 

III.  The  Tests 8 

Limitations  as  to  the  Subject  Matter 8 

Aim  in  Selecting  Exercises  and  Difficulties 

Involved 8 

The  Preliminary  Tests 9 

Final  Selection  of  Exercises 10 

Description  of  Tests 10 

IV.  Giving  the  Tests 19 

Class  of  Pupils  Tested 19 

Time  when  the  Tests  were  Given 19 

Schools  in  which  the  Tests  were  Given ....  20 
Means  of  Securing  Uniformity  in  Giving  the 

Tests 21 

V.  Scoring  the  Papers 24 

General  Statement  of  Method 24 

Means  of  Securing  Uniformity  of  Scoring .  .  28 

Scoring  each  Test 28 

Test  A 28 

V 


VI  TABLE  OF   CONTENTS 

Test  B 32 

TestC 36 

Test  D 38 

Test  E .  . 42 

Weighting  the  Exercises 43 

VI .  Critical  Examination  of  the  Tests 49 

VI I .  Examination  of  School  Grades 59 

VIII.  Comparison  of  Test  and  School  Grades 64 

Method  of  Determining  the  Correlation ...  64 
Method   of   Dealing  with   the   Data   from 

Different  Schools 66 

The  Coefficients  of  Correlation 66 

Conclusions 72 

IX.  The  Extent  to  which  the  Abilities  are  Developed  72 

Constancy  of  Results 72 

Standards  of  Achievements 74 

Conclusions 93 

X.  Use  of  the  Tests 94 

XI.  Conclusions 95 

Appendix. 

I.  A    Brief   Statement   of    Directions    for   Scoring 

Papers 97 

Test  A 97 

Test  B .  .  .    98 

Test  C 98 

Test  D 99 

Test  E 99 

II.  Information  Blank 99 

III.  The  Text  Book 100 

IV.  The  Amount  of  Time  Given  to  the  First  Two 

Books  of  Geometry 101 

V.  The  Place  of  Geometry  in  the  Curriculum 104 

VI.  The  Amount  of  Time  Given  to  Algebra  before 

beginning  the  Study  of  Geometry 106 

VII.  Experimental  and  Constructional  Geometry.  ...  106 

VIII.  The  Methods  Used.  .                                              .  106 


INTRODUCTION 


In  1911-12  the  author  had  charge  of  about  forty  pupils  in 
second-term  plane  geometry  in  the  Bloomington,  Indiana,  high 
school.  Most  of  these  pupils  came  from  good  homes  and 
apparently  should  have  been  able  to  do  satisfactory  work. 
Such,  however,  was  not  the  case.  Crudely  constructed  tests 
revealed  almost  complete  lack  of  certain  abilities  believed  to  be 
fundamental  to  the  study  of  geometry.  Special  attention  given 
to  these  abilities  resulted  in  a  decided  improvement  in  the  work 
of  the  group.  Since  then  the  author  has  experimented  along 
these  same  lines  with  other  classes  and  in  the  light  of  this  experi- 
ence it  seemed  desirable  to  determine  the  relation  of  these  specific 
abilities  to  general  geometrical  ability.  During  the  years  1915- 
17  he  conducted  an  extensive  investigation  which  is  reported  in 
the  following  pages. 

So  far  as  the  author  knows,  no  similar  investigation  has  been 
carried  on  in  this  field.  L.  V.  Stockard  and  J.  Carl  ton  Bell1 
have  made  "A  Preliminary  Study  of  the  Measurement  of  Abili- 
ties in  Geometry,"  but  it  does  not  cover  the  ground  of  this 
investigation.  Suggestions  as  to  methods  of  procedure  and  of 
handling  the  data  have  been  gathered  from  the  following: 
Thorndike's  "Mental  and  Social  Measurements,"  Brown's 
"Mental  Measurement,"  King's  "Elements  of  Statistical 
Methods,"  Buckingham's  "Spelling  Ability — Its  Measurement 
and  Distribution,"  Stone's  "Arithmetical  Abilities,  Some  Factors 
Determining  Them,"  Trabue's  "Completion-Test  Language 
Scales,"  and  Woody's  "Measurements  of  Some  Achievements  in 
Arithmetic." 

This  report  consists  of  two  parts.  The  first  is  a  brief  synopsis 
of  the  method  and  results  and  is  intended  for  those  who  care 
only  for  the  conclusions ;  the  second  is  a  more  detailed  statement 
including  the  data,  and  is  intended  for  those  who  care  to  investi- 
gate the  study  more  carefully. 

1  Journal  of  Educational  Psychology,  Vol.  7,  pp.  567-580. 

vii 


AN  INVESTIGATION  OF  CERTAIN  ABILITIES 

FUNDAMENTAL  TO   THE  STUDY 

OF  GEOMETRY 


PART   I 

Purpose  of  the  Investigation. — Success  in  the  formal  demon- 
stration of  a  theorem  of  geometry  is  dependent  upon  at  least 
four  abilities;1  namely, 

1.  The  ability  to  draw  a  figure  for  the  theorem, 

2.  The  ability  to  state  concretely  and  accurately  the  hypothesis 
and  conclusion  of  the  theorem, 

3.  The  ability  to  recall  additional  facts  about  a  figure  when 
one  or  more  facts  are  given,  and 

4.  The  ability  to  select  from  the  available  facts  those  that  are 
necessary  for  a  proof  and  to  arrange  them  so  as  to  arrive  at  the 
desired  conclusion. 

The  purpose  of  this  investigation  is  threefold : 

1.  To  determine  the  relation  of  each  of  these  four  abilities  to 
the  teachers'  marks.     This  should,  in  turn,  determine  either  the 
extent  to  which  teachers  value  these  abilities  or  the  degree  to 
which  they  are  able  to  base  their  marks  on  the  things  which  they 
do  value. 

2.  To  determine  the  extent  to  which  these  abilities  are  de- 
veloped in  our  high  schools. 

3.  To  develop  tests  which  may  be  used  for  the  purpose  of 
diagnosis;    that  is  for  the  purpose  of  determining  whether  or 
not  the  weaknesses  of  a  class  are  due  to  the  lack  of  development 
of  one  or  more  of  these  abilities. 

Plan  of  Procedure. — For  these  purposes  four  tests2  (to  be 
known  as  A,  B,  C,  and  D)  have  been  developed,  each  testing  one 
of  the  abilities  in  question.  A  fifth  test  (E)  requiring  pupils  to 
draw  the  auxiliary  lines  for  exercises  was  developed  to  supple- 

1  Pages  6-7. 

2  Pages  10-19. 


2  INVESTIGATION   OF  CERTAIN 'ABILITIES 

ment  those  testing  abilities  3  and  4  above.  In  order  to  avoid 
the  effect  of  extensive  class  drill,  original  exercises  were  selected 
for  each  of  these  tests.  None  of  the  exercises  involved  concepts 
or  knowledge  beyond  the  first  two  books  of  geometry. 

Each  test  was  given  to  at  least  one  thousand  pupils,  and  after 
the  papers  were  carefully  marked  the  coefficient  of  correlation 
between  the  test  scores  and  the  school  grades  was  computed  for 
each  school  tested.  These  coefficients  serve  as  indices  of  the 
relation  between  the  pupils'  abilities  and  the  teachers'  marks. 
The  median  scores  for  each  test  have  been  determined  and  they 
will  serve  as  standards  for  determining  whether  or  not  a  class  is 
weak  in  respect  to  any  one  of  these  abilities. 

Giving  the  Tests. — The  tests  were  given  in  sixty-three  schools 
in  the  states  of  New  York,  Rhode  Island,  New  Jersey,  Penn- 
sylvania, West  Virginia,  Ohio,  Indiana,  Illinois,  Iowa  and 
Minnesota.  Each  test  was  given  to  at  least  one  thousand 
pupils,  and  in  all  5,195  pupils  were  tested.  Among  these  was 
included  almost  every  type  of  pupil.  The  tests  were  given  when 
the  classes  had  completed  the  first  two  books  of  geometry,  thus 
avoiding  the  extensive  elimination  of  pupils  which  occurs  later 
in  the  course.  This  procedure  is  further  justified  by  the  fact 
that  the  abilities  with  which  this  study  is  concerned  should  be 
developed  to  a  considerable  degree  by  this*  time,  and  for  the 
purpose  of  diagnosis  it  is  important  that  the  tests  should  be 
given  as  early  as  possible.  The  widj^Bpibution  of  schools 
made  it  impossible  for  the  author  to  C(|I|V^  the  tests  in  person. 
Therefore,  the  teacher  of  each  class  tested  h$r  own  pupils.  To 
insure  uniformity  a  simple  but  definite  set  of  directions  was 
sent  to  each  examiner.1 

Scoring  the  Papers. — Two  scores  were  kept  for  each  pupil. 
The  one,  to  be  known  as  the  positive  score,  is  based  on  the  per 
cent,  of  necessary  statements  correctly  given;  and  the  other, 
to  be  known  as  the  negative  score,  is  the  total  number  of  in- 
correct and  unnecessary  statements  in  his  paper.  We  know  of 
no  way  of  combining  these  two  elements ;  and  as  there  is,  by  no 
means,  a  perfect  correlation2  between  the  two  scores  neither 
can  be  neglected  on  the  ground  that  the  other  gives  a  perfect 

1  Pages  21-24. 

2  Pages  25-26. 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY         3 

representation  of  the  pupil's  ability.  Furthermore,  for  the 
purpose  of  diagnosis  it  is  important  that  we  have  the  analytic 
view  given  by  the  separate  scores  rather  than  that  given  by  a 
combination  of  the  results.  Hence  for  our  purpose  it  seems 
best  to  keep  the  two  scores  separately. 

In  order  to  secure  uniformity  the  author  marked  all  papers. 
Before  beginning  to  mark  the  papers  each  exercise  of  a  test  was 
carefully  solved  and  the  number  of  necessary  steps  noted  as  a 
basis  for  computing  the  positive  scores.  Each  exercise  of  a 
test  was  then  scored  separately  for  an  entire  school ;  and  if  any 
answer  gave  particular  difficulty,  a  record  of  it  was  kept  for 
reference  in  all  similar  cases.  After  the  papers  of  all  pupils 
taking  a  given  test  were  thus  marked  the  exercises  of  that  test 
were  weighted  according  te-  the  average  positive  scores.1  A 
pupil's  final  positive^core  was  obtained  by  marking  each  exercise 
on  the  basis  of  the  weighted  value  thus  assigned  and  then  adding 
these  marks  for  all  exercises  of  the  test. 

Due  to  the  great  variation  in  the  nature  of  the  incorrect  and 
unnecessary  statements  it  was  not  possible  to  weight  each  error 
separately,  and  as  there  was  no  upper  limit  to  the  number  of 
such  statements  that  could  be  made  we  had  no  basis  for  com- 
paring the  negative  difficulties  of  two  exercises.2  Hence  the 
exercises  were  not  weighted  according  to  the  negative  scores. 
The  total  number  of  incorrect  and  unnecessary  statements  in  a 
pupil's  paper  was  taken  as  his  negative  score. 

Criticism  of  the  Tests. — If  the  time  available  for  giving  the 
tests  had  permitted  a  larger  number  of  exercises  to  be  used, 
more  satisfactory  results  would,  no  doubt,  have  been  obtained. 
An  examination  of  the  data3  shows  that  the  interval  of  difficulty 
covered  by  the  exercises  of  each  test  is  too  small  and  that  the 
exercises  are  not  distributed  uniformly  throughout  that  interval. 
In  general,  the  distribution  curves4  given  by  either  the  positive 
or  negative  scores  are  skewed  toward  the  high  end  of  the  scale, 
but  this  skewness  would,  perhaps,  largely  disappear  if  the  two 
scores  could  be  combined.  Furthermore,  since  we  are  concerned 
chiefly  with  the  ranks  of  pupils  when  arrayed  according  to  their 

1  Pages  43-49- 

2  Page  44. 

3  Pages  51-52. 
*  Pages  53-58. 


4  INVESTIGATION   OF   CERTAIN   ABILITIES 

abilities,  it  is  not  so  important  that  the  test  show  the  exact 
difference  between  the  abilities  of  two  pupils  as  it  is  that  they 
show  which  of  the  two  is  the  better.  That  Tests  A,  B,  C  and  D 
satisfy  this  condition  is  shown  by  the  fact  that  there  is  but  a 
slight  tendency  to  group  the  pupils  about  a  few  points  of  the 
scale.  However,  in  Test  E  the  pupils  are  for  the  most  part 
grouped  at  only  four  points.1  Hence  we  may  conclude  that  for 
our  purpose  Tests  A,  B,  C  and  D  are  fairly  satisfactory  but  that 
Test  E  is  not. 

Criticism  of  Teachers'  Grades. — An  examination  of  the  grades 
given  by  the  teachers  to  the  5,195  pupils  tested  shows  that  the 
grades  of  a  large  number  of  schools  taken  together  give  a  fairly 
normal  distribution  curve.2  If,  however,  the  grades  of  the  schools 
are  considered  separately,  there  is  a  great  variation  in  the  form 
of  the  distribution.3  Occasionally  the  curve  of  a  school  is 
remarkably  normal,  but  usually  it  presents  some  marked  irregu- 
larity. Hence  we  may  conclude  that  usually  teachers'  grades 
are  not  reliable  measures  of  pupils'  abilities. 

Coefficients  of  Correlation. — The  coefficient  of  correlation  be- 
tween the  test  and  school  grades  has  been  computed  for  each 
school  tested.4  This  coefficient  varies  from  —  0.150  to  0.697 
for  Test  A;  from  0.140  to  0.588  for  Test  B;  from  0.042  to  0.548 
for  Test  C;  from  0.126  to  0.568  for  Test  D;  and  from  0.139  to 
0.608  for  Test  E.  In  the  case  of  each  test  the  coefficient  is  usually 
small.  If  the  abilities  which  this  study  investigates  are  of  value 
in  themselves,  or  if  they  form  the  basis  for  other  results  which 
are  of  value,  they  should  bear  a  closer  relation  to  the  school 
grades  than  these  coefficients  indicate.  If,  on  the  other  hand, 
the  coefficients  of  correlation  can  be  taken  as  indices  of  the 
values  of  these  abilities,  then  these  values  are  so  slight  that  the 
schools  are  scarcely  justified  in  giving  as  much  time  to  this 
phase  of  geometry  as  is  now  given  to  it. 

Standards  of  Achievements. — The  median  positive  scores5  for 
the  different  tests  are:  Test  A,  62.5;  Test  B;  69.3;  Test  C, 
50.6;  Test  D,  73.3;  and  Test  E,  61.5.  The  median  negative 

1  Tables  XXVII-XXXVII. 

*  Page  59- 

1  Pages  61-63. 

*  Tables  XVIII  XXII. 
4  Page  94. 


FUNDAMENTAL   TO   THE    STUDY  OF   GEOMETRY  5 

scores  are:  Test  A,  7.1;  Test  B,  3.5;  Test  C,  4.1;  Test  D,  2.6; 
and  Test  E,  2.5.  A  study  of  the  data  from  each  school  shows 
that  in  the  case  of  each  test,  the  marks  of  some  of  the  schools 
are  quite  satisfactory  while  those  of  others  are  extremely  low. 
Local  conditions  have,  no  doubt,  tended  to  lower  the  marks  of 
some  schools;  nevertheless,  it  is  difficult  to  see  why  the  results 
should  be  so  poor  in  some  cases.  If  the  abilities  tested  are 
essential  to  success  in  the  study  of  geometry,  then  the  results 
indicate  that  progress  in  some  schools  is  almost  impossible  until 
these  abilities  have  been  further  developed.  On  the  other  hand 
the  achievements  of  other  schools  indicate  that  it  is  altogether 
possible  to  develop  these  abilities  to  a  fair  degree  during  the 
study  of  the  first  two  books  of  geometry. 


PART   II 
I.  PURPOSE  OF  THE  INVESTIGATION 

High  school  geometry  should  include  both  the  formal  and  the 
practical  phases  of  the  subject.  The  most  important  parts  of 
formal  geometry  are  the  demonstration  of  theorems,  the  con- 
struction of  figures  under  given  conditions,  and  the  solution  of 
numerical  problems.  This  study  is  limited  to  an  investigation 
of  certain  fundamental  abilities  involved  in  the  demonstration 
of  theorems. 

An  examination  of  the  steps  in  a  demonstration  will  reveal 
these  abilities.  The  first  step  in  a  demonstration  is  to  draw  the 
figure  described  in  the  theorem.  As  a  second  step,  the  pupil 
should  state  the  hypothesis  and  conclusion  accurately  in  terms 
of  his  figure.  The  third  step  is  to  recall  additional  known  facts 
concerning  the  figure.  Only  in  the  simplest  cases  will  the  con- 
clusion follow  directly  and  solely  from  the  facts  stated  in  the 
hypothesis.  Hence  it  is  necessary  for  the  pupil  to  have  the 
important  properties  of  a  geometrical  figure  so  definitely  asso- 
ciated that  he  can  recall  them  at  will.  Finally,  as  a  fourth 
step,  it  is  necessary  to  select  from  all  the  available  facts  those 
essential  to  the  proof  and  to  arrange  them  in  the  order  necessary 
to  arrive  at  the  desired  conclusion.  Here  there  are  really  two 
steps  involved, — the  selection  and  the  arrangement  of  facts. 
However,  these  steps  are  so  closely  related  that  it  seems  im- 
possible to  separate  them  for  the  purposes  of  this  investigation. 
The  selection  of  facts  to  be  used  will  depend  upon  the  arrange- 
ment or  the  method  of  proof.  On  the  other  hand  the  method 
of  proof  must  depend  upon  the  facts  which  can  be  recalled. 
Hence  in  this  study  these  two  elements  are  considered  as  a  single 
step.  Corresponding  to  these  four  steps  are  the  four  funda- 
mental abilities  with  which  this  study  is  concerned ;  namely, 

1.  The  ability  to  draw  a  figure  for  a  theorem, 

2.  The  ability  to  state  the  hypothesis  and  conclusion  accurately 
in  terms  of  the  figure, 

3.  The  ability  to  recall  additional  known  facts  concerning  the 
figure,  and 

6 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY         7 

4.  The  ability  to  select  the  necessary  facts  and  to  arrange 
them  so  as  to  produce  a  proof. 

The  purpose  of  this  investigation  is  threefold : 

1.  To  determine  the  relation  of  each  of  these  four  abilities  to 
the  teachers'  marks.     This  should,  in  turn,  determine  either  the 
extent  to  which  teachers  value  these  abilities  or  the  degree  to 
which  they  are  able  to  base  their  marks  on  the  things  which  they 
do  value. 

2.  To  determine  the  extent  to  which  these  abilities  are  de- 
veloped in  our  high  schools. 

3.  To  develop  tests  which  may  be  used  for  the  purpose  of 
diagnosis;    that  is  for  the  purpose  of  determining  whether  or 
not  the  weaknesses  of  a  class  are  due  to  the  lack  of  development 
of  one  or  more  of  these  abilities. 

II.  BRIEF  STATEMENT  OF  PLAN  OF  PROCEDURE 

It  seems  reasonable  to  suppose  that  school  grades  are  measures 
of  the  abilities  essential  to  the  particular  kind  of  work  accepted 
by  teachers  as  indicating  a  successful  mastery  of  their  subjects. 
If  the  teacher  holds  the  pupils  for  original  demonstrations  and  if, 
as  we  believe,  the  four  abilities  enumerated  above  are  essential 
to  such  work,  then  the  teacher's  grades  will,  to  a  certain  extent, 
be  measures  of  these  abilities.  Hence  if  we  can  measure  each 
of  these  abilities  separately,  the  results  should  bear  a  definite 
relation  to  the  teacher's  grades.  For  this  purpose  four  tests 
(to  be  referred  to  as  Test  A,  B,  C  and  D1)  have  been  arranged. 
Test  E1  has  been  used  to  supplement  the  other  tests,  especially 
Tests  C  and  D.  In  the  case  of  each  test  all  elements  except  the 
one  tested  for  have  been  eliminated  so  far  as  it  was  possible. 
Each  test  was  given  to  at  least  one  thousand  pupils  in  various 
schools  in  this  country,  and  the  coefficients  of  correlation  between 
the  pupils'  test  scores  and  their  school  grades  have  been  com- 
puted. These  coefficients  have  been  taken  as  indices  of  the 
extent  to  which  these  abilities  influence  the  teacher's  grades. 
As  standards  to  be  used  for  the  purpose  of  comparison  the 
median  score  has  been  computed  for  each  test. 

1  The  attention  of  teachers  who  gave  Test  D  or  Test  E  to  their  pupils  is  called 
to  the  fact  that  the  letters  designating  these  two  tests  have,  for  the  sake  of  logical 
order,  been  interchanged.  If  any  teacher  gave  Test  D  she  will  find  the  data  re- 
corded under  Test  E  and  vice-versa. 


8  INVESTIGATION   OF   CERTAIN   ABILITIES 

III.  THE  TESTS 

Limitations  as  to  the  Subject  Matter. — Because  of  their  more 
extensive  knowledge  of  the  subject  it  would  be  desirable  to  include 
in  such  a  study  those  pupils  who  have  completed  all  of  plane 
geometry.  But  with  this  advantage  there  would  come  certain 
disadvantages.  First,  if  we  should  include  these  more  advanced 
pupils  it  would  be  necessary  to  eliminate  those  who  have  com- 
pleted only  two  books  of  geometry,  unless  a  second  set  of  ques- 
tions should  be  used  for  this  group,  but  this  would  unduly 
increase  the  work  of  the  investigation.  Second,  if  we  should 
thus  limit  the  investigation  to  pupils  having  recently  completed 
all  of  plane  geometry,  we  would  have  a  very  specially  selected 
group,  since  extensive  elimination  has  taken  place  before  this, 
and  our  data  would  not  be  representative.  Third,  the  group  of 
pupils  having  completed  the  whole  of  plane  geometry  is  com- 
paratively small  and  the  difficulty  in  securing  the  desired  number 
would  be  increased.  On  the  other  hand,  it  seems  reasonable  to 
suppose  that  by  the  time  a  class  has  completed  the  first  two 
books  of  geometry  the  abilities  in  question  will  be  sufficiently 
developed  to  play  an  important  part  in  the  study  of  the  subject 
and  to  bear  a  definite  relation  to  the  teacher's  grades  if  these 
abilities  are  fundamental  to  the  kind  of  work  which  she  demands. 
Also  if  the  tests  are  to  be  used  for  the  purpose  of  diagnosis,  it  is 
necessary  that  they  should  be  given  before  the  study  of  geometry 
is  completed.  Hence  the  tests  have  been  limited  to  the  subject 
matter  included  in  the  first  two  books  of  geometry. 

Aim  in  Selecting  Questions  and  Difficulties  Involved. — In 

arranging  the  tests  our  aim  has  been  to  satisfy  the  following 
conditions :  The  tests  shall  include  only  such  exercises  as  require 
a  knowledge  of  the  first  two  books  of  geometry  for  their  solution. 
The  material  shall  be  as  varied  and  as  inclusive  as  possible. 
The  exercises  shall  vary  in  degree  of  difficulty,  proceeding  by 
more  or  less  equal  intervals  from  a  comparatively  easy  exercise 
to  one  which  offers  considerable  difficulty.  Original  exercises 
only  shall  be  included,  thus  avoiding,  as  nearly  as  possible, 
propositions  which  have  received  special  attention  in  class. 
Each  test  shall  be  of  such  length  that  it  can  be  given  in  a  recita- 
tion period  without  involving  the  speed  factor. 

Certain  difficulties  have  been  met  in  the  attempt  to  attain 


FUNDAMENTAL  TO  THE   STUDY  OF  GEOMETRY  9 

these  ideals.  First,  the  nature  of  the  material  has  made  it 
impossible  to  include  a  large  number  of  exercises  in  the  pre- 
liminary tests,1  and  our  field  of  choice  has  therefore  been  limited. 
Tests  in  arithmetic,  spelling,  composition,  and  other  subjects 
are  of  such  a  nature  that  it  is  possible  to  give  a  large  number  of 
exercises  in  a  few  minutes.  It  is  then  possible  to  make  a  more 
satisfactory  selection  than  we  have  been  able  to  make.  Also 
the  number  of  high  school  pupils  of  the  desired  grade  is  com- 
paratively small,  and  the  course  in  geometry  is  so  extensive 
that  teachers  are  seldom  justified  in  giving  more  than  one  period 
to  such  an  investigation,  thus  making  it  difficult  to  secure  a  large 
group  for  the  preliminary  tests  or  to  repeat  a  test  if  the  first  data 
failed  to  be  satisfactory. 

The  Preliminary  Tests. — From  various  texts  five  sets  of  thirty 
exercises  each  were  selected  on  the  basis  of  their  probable  fitness 
for  the  particular  test  in  question.  These  exercises  were  carefully 
solved  and  ten  were  selected  from  each.  Each  set  of  ten  exercises 
was  arranged  in  a  manner  similar  to  that  described  on  pages 
10-19  and  directions  for  giving  the  tests  were  formulated.  Each 
test  was  then  given  to  about  thirty  pupils  in  a  representative 
school  in  order  to  determine,  as  far  as  possible,  whether  any 
changes  should  be  made  in  the  arrangement  of  the  tests,  in  the 
wording  of  the  exercises,  in  the  number  of  exercises  in  each  test, 
or  in  the  directions  for  giving  the  tests.  As  a  result  of  this  trial 
the  number  of  exercises  in  each  test  was  reduced  and  the  direc- 
tions for  giving  Test  D  were  revised. 

During  the  second  half  of  the  school  year  1915-16  each  revised 
test  was  given  to  about  two  hundred  pupils  in  order  to  secure 
data  for  the  final  selection  of  exercises.  To  avoid  specially 
selected  groups,  it  was  desirable  to  give  these  preliminary  tests 
to  as  great  a  variety  of  pupils  as  possible.  Hence  they  were 
given  in  thirteen  schools  in  the  states  of  New  York,  New  Jersey, 
Pennsylvania,  West  Virginia,  Indiana,  Illinois  and  Iowa.  To 
guard  further  against  a  single  test  being  given  to  a  large  number 
of  pupils  working  under  the  same  special  conditions,  two  or 
more  tests  were  given  in  each  of  the  larger  schools.  This  gave 
for  each  test  data  gathered  from  several  groups  of  pupils  working 
under  varied  conditions. 

1  See  Preliminary  Tests  below. 


IO  INVESTIGATION   OF   CERTAIN   ABILITIES 

The  wide  distribution  of  schools  made  it  impossible  for  the 
author  personally  to  conduct  the  tests  in  each  school,  although 
he  did  so  in  those  schools  which  were  near  at  hand.  However, 
the  simplicity  of  the  tests  made  it  possible  to  prepare  directions 
which  were  simple  and  definite,  and  the  results  indicate  that 
there  has  been  no  material  variation  from  these  directions. 
Two  sets  of  instructions  were  sent  to  each  school.  One  gave 
directions  for  certain  preliminaries  and  for  the  disposal  of  the 
papers  after  the  test  had  been  given.  The  other  gave  specific 
directions  for  conducting  the  test,  These  instructions  were  the 
same  as  those  on  pages  21-24  except  that  the  examiner  was 
directed  to  note  carefully  the  time  required  by  each  pupil,  and 
there  was  no  provision  for  collecting  the  papers  after  thirty 
minutes  of  actual  work. 

Final  Selection  of  Exercises. — After  the  papers  were  carefully 
graded  the  exerciseo  for  the  final  tests  were  selected.  From  the 
experience  gained  in  giving  the  preliminary  tests  it  was  evident 
that  not  more  than  thirty  minutes  were  available  for  actual 
work  during  a  recitation  period.  The  time  spent  by  each  pupil 
and  the  amount  of  work  completed  indicated  that  not  more  than 
the  following  number  of  exercises  could  be  included  in  each  test : 
Test  A,  five;  Test  B,  four;  Test  C,  four;  Test  D,  three;  and 
Test  E,  four.  Some  pupils  will  not  require  the  full  thirty 
minutes  for  these  tests  but,  as  we  desired  to  eliminate  the  speed 
factor,  the  time  allowed  should  be  ample  for  all  pupils.  After 
thus  selecting  the  exercises  for  Test  A  and  giving  them  to  about 
three  hundred  pupils,  it  appeared  that  they  were  too  easy  to 
be  effective  and  a  new  set  of  exercises  was  selected  in  the  same 
manner  as  described  above.  The  data  from  these  preliminary 
tests  have  been  omitted  since  the  results  of  the  final  tests1  will 
serve  as  an  effective  check  on  the  validity  of  the  choice  of  ques- 
tions. 

Description  of  the  Tests. — Each  exercise  thus  finally  selected 
for  a  given  test  was  printed  on  a  separate  sheet  of  paper  with 
space  below  for  the  pupil's  answer.  These  sheets  together  with 
a  cover-sheet  were  bound  at  the  top  so  as  to  give  freedom  in 
folding  them  back  while  the  pupil  worked.  The  content  of  the 

1  Pages  49-59- 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  II 

cover-sheet  for  each  test  was  the  same  as  the  following  for  Test 
A  except  for  the  letter  designating  the  test: 

TEST  A. 

Are  you  a  boy  or  a  girl? 

Your  Name 

Town School 

Date 

Your  Teacher's  Name . . 


Test  A. 

The  purpose  of  this  test  was  to  measure  the  pupil's  ability  to 
draw  accurate  figures  for  theorems. 

The  exercises  follow  in  the  order  in  which  they  were  given : 

I 

Draw  the  figure  for  the  following  proposition : 

If  two  radii  of  a  circle  are  perpendicular,  and  a  tangent  to  the 
circle  cuts  these  radii  produced  at  points  A  and  B,  the  other 
tangents  drawn  from  A  and  B  are  parallel. 

Draw  the  figure  for  the  following  proposition : 

If  two  lines  which  are  on  opposite  sides  of  a  third  line  meet  at 
a  point  of  the  third  line,  making  the  non-adjacent  angles  equal, 
the  lines  form  one  and  the  same  line. 

Ill 

Draw  the  figure  for  the  following  proposition : 

The  perpendicular  drawn  from  the  point  of  intersection  of  the 
medians  of  a  triangle  to  a  line  without  the  triangle  is  equal  to 
one  third  the  sum  of  the  perpendiculars  from  the  vertices  of  the 
triangle  to  the  line. 

IV 

Draw  the  figure  for  the  following  proposition : 
The  bisectors  of  the  interior  and  the  exterior  vertical  angles  of 


12 


INVESTIGATION   OF   CERTAIN   ABILITIES 


a  triangle  meet  the  circumscribed  circumference  in  the  mid- 
points of  the  arcs  into  which  the  base  divides  the  circumference, 
and  the  line  joining  those  points  is  the  diameter  which  bisects 
the  base. 

V 

Draw  the  figure  for  the  following  proposition : 

The  bisectors  of  the  angles  included  between  the  opposite 
sides  (produced)  of  an  inscribed  quadrilateral  intersect  at  right 
angles. 

Test  B. 

The  purpose  of  this  te»t  was  to  determine  the  pupil's  ability 
to  state  the  hypothesis  and  conclusion  of  a  theorem  in  terms 
of  a  given  figure. 

The  exercises  follow  in  the  order  in  which  they  were  given  : 

I 

State  what  is  given  and  what  is  to  be  proved  in  the  following 
proposition  : 

If  two  circles  intersect,  the  common  secant  drawn  through 
one  of  the  points  of  intersection  and  parallel  to  the  line  of  centers 
is  greater  than  any  other  common  secant  drawn  through  that 
point  of  intersection. 


Given  :l 
To  prove: 


II 


State  what  is  given  and  what  is  to  be  proved  in  the  following 
proposition : 

If  two  circles  are  tangent  internally  and  chords  of  the  outer 

1  Ample  space  was  left  after  "Given"  and  "To  prove"  for  full  answers. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  13 

circle  are  drawn  tangent  to  the  inner  circle,  that  chord  is  greatest 
which  is  parallel  to  the  common  tangent. 


Given : 
To  prove: 


III 


State  what  is  given  and  what  is  to  be  proved  in  the  following 
proposition : 

If  from  the  extremities  of  a  given  side  of  a  triangle  perpendicu- 
lars are  drawn  to  the  bisector  of  the  angle  opposite  that  side,  the 
lines  connecting  the  feet  of  these  perpendiculars  to  the  mid- 
point of  the  given  side  are  equal,  and  either  is  equal  to  half  the 
difference  of  the  other  two  sides  of  the  triangle. 


Given : 
To  prove: 


IV 


State  what  is  given  and  what  is  to  be  proved  in  the  following 
proposition : 

An  angle  of  a  triangle  is  a  right  angle,  an  acute  angle,  or  an 
obtuse  angle,  according  as  the  median  drawn  from  the  vertex 


14  INVESTIGATION   OF   CERTAIN   ABILITIES 

of  the  angle  is  equal  to,  greater  than,  or  less  than  one  half  of 
the  opposite  side. 

C 


Given : 
To  prove: 


Test  C. 


This  test  was  arranged  to  measure  the  pupil's  ability  to  recall 
known  facts  about  figures  when  one  or  more  facts  are  given 
The  questions  follow  in  the  order  in  which  they  were  given  : 


I 


Given:  Triangle  ABC,    /I  =  a  right  angle,  and    Z3  =  two 
times  Z2. 
State  as  many  more  facts  about  the  above  figure  as  you  can. 


II 


Given:  The  square  ABCD,  the  diagonal  BD,  EB  =  CD  and 
EF  is  perpendicular  to  BD. 

State  as  many  more  facts  about  the  above  figure  as  you  can. 


FUNDAMENTAL   TO   THE    STUDY   OF  GEOMETRY 

III 
X 


Given  two  circles  0  and  0'  intersecting  in  C  and  D,  the  di- 
ameters AC  and  CB,  and  the  line  AB. 

State  as  many  more  facts  about  the  above  figure  as  you  can 


IV 


Given:  Triangle  ABC,  AE  bisects  angle  CAB,  BF  bisects 
angle  ABC,  AE  and  BF  intersect  in  0,  PR  is  drawn  through  O 
parallel  to  AB. 

State  as  many  more  facts  about  the  above  figure  as  you  can. 

Test  D. 

The  purpose  of  this  test  was  to  determine  the  pupil's  ability 
to  select  and  organize  facts  to  produce  a  proof.  At  the  top 
of  each  sheet  there  was  a  figure.  Below  was  a  statement  of 
what  was  given  and  what  was  to  be  proved.  To  eliminate  the 
factor  tested  for  by  Test  C  a  list  of  "Other  known  facts"  was 
given  at  the  left  hand  side  of  the  lower  half  of  the  sheet.  To  the 
right  of  this  list  was  ample  space  for  the  pupil's  proof.  The  list 
of  "Other  known  facts"  contained,  among  other  facts,  those 
essential  to  the  proof.  The  pupil  was  free  to  select  facts  from 
this  list  if  the  figure  did  not  suggest  them. 

The  questions  in  the  order  in  which  they  were  given  follow : 


16 


INVESTIGATION   OF   CERTAIN   ABILITIES 


Given:  AB  =  AD,  ED  is  perpendicular  to  DB,   CB  is  per- 
pendicular to  BD,  and  EC  passes  through  A. 

To  prove  that  triangle  ABC  is  congruent  to  triangle  EAD. 
Other  known  facts  :  Proof  : 

Z6  =  Z3 
Zi  =  Z4 
Z4  +  Z7  =  180 
Z2  =  Z5 
ED  is  parallel  to 


-  CB  <AB 


II 


Given:  P  is  any  point  within  the  circle  0,  A  C  is  a  diameter 
through  P,  BD  is  any  other  chord  through  P,  OD  is  a  radius. 

To  prove  that  AP  >  DP. 

Other  known  facts:  Proof: 

Z6  =   Z3  +  Z4 
PD  -  OD  <  OP 
OD  +  OP  >  DP 
Z2  =   Z3 
AO  = 


Z5  +  Z6  =  180° 
AO  +  OP  =  AP 


FUNDAMENTAL   TO   THE    STUDY  OF  GEOMETRY 
III 


Given:  The  triangle  ABC  inscribed  in  the  circle  whose  center 
is  0,  CD  is  perpendicular  to  AB,  BE  is  perpendicular  to  AC. 

To  prove  that  arc  AE  =  arc  AD. 
Other  known  facts:  Proof: 

Z8  =  90° 

Z  6  is  measured  by  half  of  arc  BD 

Z  7  is  measured  by  half  of  arc  AD 

Z  5  is  measured  by  half  of  arc  EC 

Z4  is  measured  by  half  of  arc  AE 

Z  2  =90° 

Z9  =   Z5  +  Z6  +  Z7 
AC  <AF  +  FC 

Z9  =  90° 

AC  -  AF  <  FC 

Z2  =   Z4  +  Z5  +  /6 
Z2  =   Z8 

Test  E. 

The  purpose  of  this  test  was  to  add  further  evidence  to  the 
.results  of  Tests  C  and  D  by  determining  the  pupil's  ability 
to  draw  auxiliary  lines.  It  is  assumed  that  if  a  pupil  can  draw 
the  auxiliary  lines  necessary  for  the  proof  of  a  theorem  he  must 
have  in  mind  a  definite  proof  and  the  facts  necessary  to  that  proof. 
The  validity  of  this  assumption  will  be  discussed  later.1  How- 
ever, if  this  is  a  true  assumption  a  test  requiring  the  pupils  to 
draw  the  lines  necessary  to  the  proof  of  a  theorem  is  a  measure 
of  his  ability  to  recall  facts,  and  select  and  arrange  them  to  pro- 
duce the  proof. 

The  questions  follow  in  the  order  in  which  they  were  given : 

1  Page  50. 


18 


IiXVESTIGATION   OF   CERTAIN   ABILITIES 
I 


•D 


Given  :  AB  is  parallel  to  CD,  and  lines  GE  and  FE  meet  in  E. 
Make  any  additional  drawings  that  are  necessary  to  prove  that 
Z2  =  /i  +  Z3. 


II 


Given:  The  circle  whose  center  is  0,  arc  AB  =  arc  BC,  BM  is 
perpendicular  to  AO,  BN  is  perpendicular  to  OC. 

Make  any  additional  drawings  that  are  necessary  to  prove  that 
BM  =  BN. 


Ill 


Given:  Triangle  ABC  inscribed  in  a  circle  whose  center  is  0, 
and  OD  perpendicular  to  CB. 

Make  any  additional  drawings  that  are  necessary  to  prove  that 
Zi  =  Z2. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  19 

IV 


D 


Given:  Circles  0  and  0'  tangent  externally  at  A,  EC  and  DE 
drawn  through  the  point  of  tangency  and  terminating  in  the 
circumferences. 

Make  any  additional  drawings  that  are  necessary  to  prove 
that  BE  is  parallel  to  DC. 

IV.  GIVING  THE  TESTS 

Class  of  Pupils  Tested. — For  reasons  stated  on  page  8  only 
those  pupils  who  had  recently  completed  the  first  two  books  of 
geometry  were  tested.  Some  teachers  proposed  giving  the  tests 
to  small  groups  of  their  brighter  pupils.  In  order  to  avoid  such 
a  specially  selected  group,  all  pupils  of  a  school  completing  the 
required  work  at  a  given  time  took  the  same  test.  It  was  not 
possible  to  give  all  the  tests  in  the  same  school  for  this  would 
have  required  five  days. 

Since  all  types  of  pupils  should  be  included,  it  was  desirable 
to  give  the  tests  only  to  classes  in  which  there  had  been  no 
elimination.  This  was  not  always  possible  without  seriously 
delaying  the  investigation.  Some  schools  spend  more  than  one 
semester  on  the  first  two  books  of  geometry  and  in  such  schools 
some  elimination  had  usually  taken  place  before  the  required 
work  had  been  completed.  If  these  schools  had  been  excluded, 
the  number  of  available  pupils  would  have  been  reduced  and  the 
difficulty  in  securing  the  desired  number  greatly  increased. 
Hence  such  schools  have  been  included.  The  probable  effect 
of  the  resulting  elimination  will  be  discussed  in  connection 
with  the  data.1 

Time  when  the  Tests  were  Given. — The  tests  could  not  well 
be  given  at  the  close  of  the  year  1915-16  for,  as  indicated  above, 

'Page  55. 


2O  INVESTIGATION   OF   CERTAIN   ABILITIES 

in  many  schools  the  pupils  who  began  the  study  of  geometry 
at  the  middle  of  that  year  did  not  complete  the  required  two 
books  until  the  following  year.  Hence  it  was  decided  to  give 
the  tests  in  the  fall  of  1916  to  pupils  who  completed  the  required 
work  at  that  time  or  at  the  close  of  the  preceding  year;  in 
January,  February  and  March  of  1917  to  pupils  who  completed 
the  work  about  that  time;  and  again  at  the  close  of  the  school 
year  1916-17.  In  order  to  overcome  the  effect  of  the  long 
summer  vacation  the  classes  which  completed  the  work  in  the 
spring  of  1916  did  not  take  the  tests  until  they  had  continued 
their  study  of  geometry  for  about  a  month  in  the  fall  of  1916. 

Schools  in  which  the  Tests  were  Given. — In  order  to  guard 
further  against  a  special  selection  of  pupils  each  test  was  given 
in  several  schools  varying  as  much  as  possible  in  their  nature 
and  location.  In  all,  the  final  tests  were  given  in  sixty-three 
schools.  The  Roman  numerals1  from  I  to  LXIII  have  been 
assigned  to  these  schools,  and  throughout  the  discussion  reference 
to  a  school  will  be  by  the  number  assigned  to  it.  The  geo- 
graphical distribution  of  the  schools  follows : 

New  York:  Schools  I,  II,  III,  VIII,  IX,  XVIII,  XX,  XXVI, 
XXVII,  XXXI,  XXXIII,  XXXV,  XLI,  XLII,  XLIV,  XLV, 
XLVI,  XLVII,  XLVIII,  XLIX,  LIV,  LXIII. 

Rhode  Island:  School  XXI. 

New  Jersey:  Schools  VII,  XXXVII,  LVII. 

Pennsylvania:  Schools  VI,  XII,  XIII,  XIV,  XV,  XIX,  XXII, 
XXV,  XXVIII,  XXIX,  XXXII,  XXXIV,  XL,  XLIII,  L,  LI, 
LIII,  LV,  LIX,  LX,  LXI. 

West  Virginia:  School  XXXIX. 

Ohio:  School  LVI II. 

Indiana:  Schools  IV,  X,  XI,  XVI,  XVII,  XXX,  XXXVI, 
LII,  LVI. 

Illinois:  Schools  V,  XXXVIII,  LXII. 

Iowa:  School  XXII I. 

Minnesota:  School  XXIV. 

The  schools  in  which  each  test  was  given  will  be  indicated  in  the 
tables  containing  the  data.2 

1  The  principal  or  the  head  of  the  department  of  mathematics  may  obtain  the 
number  of  his  school  by  addressing  the  author. 
1  Pages  46-49. 


FUNDAMENTAL   TO   THE   STUDY   OF   GEOMETRY  21 

Means  of  Securing  Uniformity  in  Giving  the  Tests. — In  most 
investigations  of  this  nature  uniformity  requires  that  the  same 
person  conduct  the  test  in  all  schools.  The  wide  distribution 
of  the  schools  tested  made  this  impossible.  However,  as  indi- 
cated on  page  10,  it  was  possible  to  formulate  simple  and  definite 
directions  which  insured  a  fair  degree  of  uniformity.  The  author 
is  personally  acquainted  with  many  of  the  teachers  who  gave 
the  tests  and  knows  that  they  can  be  relied  upon  to  follow 
directions  carefully.  Letters  of  inquiry  written  by  examiners 
before  the  tests  were  given  and  comments  often  returned  with 
the  papers  indicated  the  examiner's  desire  to  follow  directions. 
In  only  two  cases  did  the  returns  from  any  of  the  schools  indicate 
a  variation  from  the  directions  and  the  papers  from  these  schools 
were  rejected. 

The  directions  for  giving  the  tests  were  as  follows : 

Instructions  to  the  Examiner. 

1.  Before  going  to  class  read  the  " Class  Rules"  carefully  and 
then  during  the  test  follow  them  without  any  variation. 

2.  On  the  day  preceding  the  test  remind  the  pupils  to  bring 
pencils,  rulers,  and  compasses  to  the  test,  unless  such  instru- 
ments are  kept  in  the  class  room. 

3.  Do  not  give  any  one  of  the  tests  unless  you  have  thirty 
minutes  for  actual  work  by  the  class. 

4.  Fill  out  one  of  the  enclosed  grade  cards  for  each  class 
tested.     Give  the  names  of  the  pupils  in  the  class  and  the  final 
school  grade  which  each  received  in  the  first  two  books  of  ge- 
ometry.    If  the  final  grades  have  not  been  made  out  retain  the 
grade  card  until  they  have  and  then  send  the  grades  to  me. 
Do  not  fail  to  send  these  grades  as  all  other  data  will  be  useless 
without  them. 

5.  When  all  the  papers  of  a  class  have  been  returned  to  you 
place  the  grade  card  for  that  class  on  top  of  the  papers  and  bind 
them  together  with  the  elastics.     Do  not  roll  the  papers;   keep 
them  in  a  flat  package. 

6.  Do  not  grade  the  papers.     Look  over  them  if  you  care  to 
see  what  your  pupils  were  able  to  do  with  the  exercises  but  do 
not  place  any  marks  on  them. 

7.  When  you  have  finished  giving  the  test  in  the  school  return 
the  papers  (Collect,  by  express)  to  J.  H.  Minnick,  81 1  N.  4Oth  St., 
Philadelphia,  Pa. 


22  INVESTIGATION    OF   CERTAIN   ABILITIES 

Class  Rules  for  Test  A 

1.  See   that  each   pupil   is   supplied   with   pencil,   ruler  and 
compass. 

2.  Read  to  the  Pupils. — I  am  going  to  give  you  some  geometry 
exercises.     In  order  that  all  of  you  may  have  the  same  chance 
I  want  you  to  start  at  the  same  time.     Do  not  open  the  set  of 
questions  which  you  are  about  to  receive  until  I  give  the  signal 
to  begin  work  by  tapping  on  the  desk. 

3.  Distribute  the  questions. 

4.  Have  pupils  fill  out  blanks  on  the  cover  sheet  of  the  ques- 
tions. 

5.  Read  to  the  Pupils. — At  the  top  of  each  sheet  which  you 
have  received  there  is  an  exercise  from  geometry.     When  the 
signal  is  given  to  begin  work  fold  back  the  cover  sheet,  read  one 
of  the  exercises  carefully,  and  then  in  the  space  below  draw  the 
figure  for  the  exercise.     Then  read  another  exercise  and  draw  the 
figure.     Continue  in   this  manner  until  you  have  drawn   the 
figures  for  all  the  exercises.     Draw  the  figures  as  accurately  as 
possible  but  you  need  not  make  actual  constructions.     Do  not 
attempt  to  prove  the  exercises.     All  I  want  to  know  is  whether 
you  can  draw  the  figure.     You  may  do  the  exercises  in  any 
order  you  care  to.     You  wtfl  have  thirty  minutes  in  which  to 
complete  your  work. 

6.  Give  the  pupils  a  chance  to  ask  questions  concerning  the 
instructions  but  do  not  reveal  the  content  of  the  questions  by 
your  answers. 

7.  Note  the  time  and  then  give  the  signal,  thus:    "Ready," 
and  then  tap  on  the  desk  with  your  pencil. 

8.  In  the  case  of  any  irregularity  on  the  part  of  any  pupil 
during  the  test  make  a  note  on  the  cover  sheet  of  his  questions 
indicating  the  exact  nature  of  the  irregularity. 

9.  Collect  all  papers  promptly  at  the  close  of  thirty  minutes 
of  actual  work. 

The  "Class  Rules"  for  the  other  tests  were  similar  to  those  for 
Test  A.  The  first  item  was  varied  according  to  the  instruments 
needed,  and  the  fifth  item  was  varied  as  follows: 

For  Test  B.— 

5.  Read  to  the  Pupils. — At  the  top  of  each  sheet  which  you  have 
received  there  is  an  exercise  from  geometry.  Just  below  this  is 


FUNDAMENTAL   TO   THE   STUDY   OF   GEOMETRY  23 

the  figure  for  the  exercise.  When  the  signal  to  begin  work  is 
given  fold  back  the  cover  sheet,  read  one  of  the  exercises  care- 
fully, and  then  in  the  space  below  state  what  is  given  and  what 
is  to  be  proved  in  the  exercise.  Then  proceed  in  the  same  manner 
with  each  of  the  other  exercises.  Be  sure  that  you  have  stated 
fully  every  fact  that  is  given.  Do  not  attempt  to  prove  the 
exercises.  All  I  want  to  know  is  whether  you  can  tell  what  is 
given  and  what  is  to  be  proved.  You  may  do  the  exercises  in 
any  order  you  care  to.  You  will  have  thirty  minutes  in  which 
to  complete  your  work. 

For  Test  C. — 

5.  Read  to  the  Pupils. — At  the  top  of  each  sheet  which  you  have 
received  there  is  a  geometrical  figure.  Below  this  there  is  a 
statement  of  what  is  given.  When  the  signal  to  begin  work  is 
given  fold  back  the  cover-sheet,  read  carefully  what  is  given  in 
one  of  the  exercises  and  then  in  the  space  below  state  as  many 
more  facts  about  the  figure  as  you  can.  When  you  have  done 
this  proceed  in  the  same  manner  with  another  exercise.  You 
may  do  the  exercises  in  any  order  you  care  to.  You  will  have 
thirty  minutes  in  which  to  complete  your  work. 

For  Test  D.— 

5.  Read  to  the  Pupils. — At  the  top  of  each  sheet  which  you  have 
received  there  is  a  geometrical  figure.  Below  this  figure  there 
is  a  statement  of  what  is  given  and  what  is  to  be  proved.  Below 
this  and  on  the  left  hand  side  of  the  sheet  there  is  a  number  of 
other  facts  about  the  figure  some  of  which  may  guide  you  in 
proving  the  exercise.  When  the  signal  to  begin  work  is  given 
fold  back  the  cover-sheet,  read  carefully  what  is  given  and  what 
is  to  be  proved  in  one  of  the  exercises,  and  then  in  the  space  at 
the  bottom  and  to  the  right  of  the  sheet  give  a  complete  proof  of 
the  exercise,  but  you  need  not  give  reasons  or  authorities  for  the 
different  steps  in  your  proof.  Refer  to  the  facts  stated  on  the 
left  hand  side  of  the  sheet  as  much  as  you  care  to  and  use  any 
of  them  that  will  help  in  your  proof.  When  you  have  com- 
pleted the  proof  of  this  exercise  proceed  in  a  similar  way  to 
prove  the  other  exercises.  You  may  do  the  exercises  in  any 
order  you  care  to.  You  will  have  thirty  minutes  in  which  to  do 
your  work. 

For  Test  E.— 


24  INVESTIGATION   OF  CERTAIN   ABILITIES 

5.  Read  to  the  Pupils. — At  the  top  of  each  sheet  which  you 
have  received  there  is  a  geometrical  figure.  Below  this  figure 
there  is  a  statement  of  what  is  given  and  what  is  to  be  done. 
When  the  signal  to  begin  work  is  given  fold  back  the  cover-sheet, 
read  carefully  what  is  given  and  what  is  to  be  proved  in  one  of 
the  exercises,  and  then  make  any  additional  drawings  that  are 
necessary  to  prove  the  exercise.  Thus,  if  in  the  triangle  ABC, 
AC  =  BC  (place  drawing  on  the  board)  and  we  are  to  prove 
that  /.A  =  Z.B  we  may  draw  CD  to  the  mid-point  of  AB. 
When  you  have  completed  this  exercise  proceed  in  a  similar 
way  to  do  the  other  exercises.  Do  not  write  any  explanations 
on  your  papers  and  do  not  prove  the  exercises.  All  I  want  to 
know  is  whether  you  can  make  the  correct  drawings.  You  may 
do  the  exercises  in  any  order  you  care  to.  You  will  have  thirty 
minutes  in  which  to  do  your  work. 

In  an  investigation  including  so  many  widely  distributed 
schools  it  is  impossible  to  secure  complete  uniformity  of  physical 
conditions.  However,  teachers  were  asked  to  report  any  con- 
ditions which  would  influence  the  results  of  the  tests,  and  from 
these  reports  it  is  believed  that  there  were  no  disturbing  elements 
of  significance. 

V.  SCORING  THE  PAPERS 

General  Statement  of  Method. — Two  scores  were  kept  for 
each  exercise.  One  was  the  per  cent,  of  necessary  elements 
given  correctly  by  the  pupil;  the  other  was  the  number  of 
incorrect  and  unnecessary  elements  given.  The  former  will 
be  referred  to  as  the  positive  score  and  the  latter  as  the  negative 
score.  We  know  of  no  valid  method  of  combining  these. 
Teachers  do  combine  them  in  some  way  or  other.  For  example, 
if  in  stating  the  hypothesis  of  a  theorem  a  pupil  gives  every  fact 
correctly  but  also  includes  one  statement  which  is  not  given, 
some  teachers  offset  the  error  by  one  of  the  correct  statements 
and  give  the  same  grade  as  if  the  pupil  had  made  no  incorrect 
statements  but  had  omitted  one  of  the  necessary  facts.  We  do 
not  know  that  any  two  such  statements  are  together  equal  to 
no  statement  at  all,  and  any  such  combination  of  correct  and 
incorrect  statements  is  arbitrary.  It  has  been  suggested  by 
some  teachers  that  the  best  way  to  meet  this  situation  is  to 
disregard  the  negative  scores  altogether.  Such  a  procedure 
assumes  one  of  two  things;  namely,  either  that  the  positive 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


scores  alone  give  the  same  distribution  of  pupils  as  the  positive 
and  negative  scores  combined,  or  that  the  misconceptions  upon 
which  the  negative  scores  are  based  have  no  significance  so  far 
as  this  study  is  concerned.  We  shall  now  show  that  neither 
of  these  assumptions  is  true  and  that  we  therefore  can  not 
neglect  the  negative  scores. 

For  this  purpose  we  have  selected  the  data  from  schools  LI  I, 
LIV  and  LVI  in  each  of  which  Test  A  was  given.  These  schools 
present  three  rather  typical  forms  of  distribution  and  are  a  fair 
representation  of  conditions  resulting  not  only  from  Test  A  but 
from  each  of  the  other  tests.  The  48  pupils  of  school  LI  I  were 
arranged  according  to  their  positive  scores,  and  then,  as  nearly 
as  possible,  divided  into  five  equal  groups.  As  48  is  not  exactly 
divisible  by  5  some  of  the  groups  are  necessarily  larger  than 
others.  The  extreme  groups  were  made  the  smaller  and  the 
entire  arrangement  was  made  symmetrical  with  respect  to  the 
central  group.  Beginning  with  the  poorest,  these  groups  were 
numbered  from  one  to  five.  In  a  similar  way,  the  pupils  were 
divided  into  groups  according  to  their  negative  scores. 

TABLE  I. — Relation  between  the  distributions  of  the  pupils  of  school  LJl  according 
to  their  positive  and  their  negative  scores  for  Test  A . 

Negative. 
A       5  4  3  2  i         D 


•55     3 
I 


3 

5 

i 

3 

3 

I 

i 

2 

2 

5 

2 

I 

1 

I 

2 

i 

3 

3 

3 

-J 

3 

10 


B       9  10         10 


10 


Table  I  shows  the  relation  of  the  distributions  given  by  each 
of  the  two  groupings.  The  columns  of  squares  represent  the 
division  of  the  class  into  fifths  according  to  negative  scores  and 
the  rows  represent  a  like  division  according  to  positive  scores. 

3 


26 


INVESTIGATION   OF   CERTAIN  ABILITIES 


Thus  the  first  row  means  that  the  nine  pupils  constituting  the 
highest  one  fifth  of  the  class  according  to  the  positive  scores  are 
distributed  according  to  the  negative  scores  as  follows:  three 
are  in  the  highest  one  fifth,  five  are  in  the  next  highest  group, 
and  one  is  in  the  group  next  to  the  lowest.  In  a  similar  way 
Tables  II  and  III  give  the  distributions  of  the  pupils  of  schools 
LIV  and  LVI  respectively. 

TABLE  II. — Relation  between  the  distributions  of  pupils  of  School  LIV  according  to 
their  positive  and  their  negative  scores  for  Test  A . 

Negative. 
AS  4  3  2  i        D 


2 

I 

I 

I 

I 

2 

I 

2 

2 

I 

I 

I 

2 

2 

2 

44644 

B 


TABLE  III. — Relation  between  the  distributions  of  pupils  of  School  LVI  according  to 
their  positive  and  their  negative  scores  for  Test  A . 

Negative. 
AS  4  3  2  i         D 


2 

4 

2 

3 

I 

I 

4 

2 

2 

4 

2 

i 

4 

4 

2 

5 

I 

3 

2 

2 

2 

3 

2 

2 

3 

12 


B         12  13  13  13  12  C 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  27 

Clearly  a  perfect  positive  correlation  would  result  in  the  ar- 
rangement of  all  pupils  along  the  diagonal  AC,  and  a  perfect 
negative  correlation  would  give  an  arrangement  of  all  pupils 
along  the  diagonal  BD.  In  Table  I  there  is  a  decided  tendency 
to  arrange  the  pupils  along  the  diagonal  A  C,  and  therefore  there 
is  a  fairly  close  correspondence  between  the  distributions  given 
by  each  set  of  scores.  On  the  other  hand,  Table  II  shows  that 
in  School  LIV  there  is  actually  a  negative  correlation  between 
the  two  sets  of  scores.  Again,  Table  III  shows  that  in  School 
LVI  there  is  but  a  slight  tendency  to  arrange  the  pupils  along 
either  diagonal.  That  is,  there  is  little  or  no  relation  between  the 
positive  and  negative  grades  in  this  school.  Hence  we  cannot  as- 
sume that  the  positive  and  negative  scores  give  the  same  distribu- 
tion of  pupils.  Furthermore  when  we  attempt  to  determine  each 
individual's  rank  we  shall  not  be  concerned  with  group  tendencies 
but  with  individual  variations,  and  even  in  those  schools  where 
there  is  a  fairly  close  correspondence  between  the  positive  and 
negative  scores  there  is  considerable  individual  variation.  Hence 
in  no  case  can  we  neglect  the  negative  scores;  provided,  of  course, 
the  misconceptions  upon  which  they  are  based  bear  a  vital 
relation  to  our  study. 

One  purpose  of  this  study,  as  we  have  previously  stated,  is  to 
determine  to  what  extent  the  abilities  in  question  influence 
teachers'  marks.  These  marks  should  be  a  measure  of  the 
pupil's  total  ability  to  do  geometry  and  any  element  influencing 
this  total  ability  should  influence  the  teachers'  marks.  But 
unnecessary  statements  tend  to  confuse  the  pupil  and  an  in- 
correct statement  can  lead  only  to  an  incorrect  conclusion. 
Hence  these  misconceptions  do  bear  a  vital  relation  to  our  first 
purpose.  Our  third  purpose  is  to  furnish  a  means  of  educa- 
tional diagnosis.  If  then,  as  we  have  just  seen,  these  mis- 
conceptions interfere  with  the  pupil's  progress  our  investigation 
must  concern  itself  with  the  negative  scores.  Hence,  since  we 
can  not  neglect  the  negative  scores  and  since  we  know  of  no 
legitimate  method  of  combining  them  with  the  positive  scores, 
it  seemed  best  to  record  them  separately.  This  procedure  is 
further  justified  by  the  fact  that,  for  the  purpose  of  diagnosis, 
it  is  essential  that  our  results  be  as  analytic  as  possible  and  that 
standards  be  established  for  each  element  entering  into  the 
study  of  geometry  rather  than  for  groups  of  elements. 


28  INVESTIGATION    OF   CERTAIN   ABILITIES 

The  exercises  of  each  test  were  weighted  according  to  the 
average  positive  scores  for  all  pupils  taking  the  test.1  Each 
pupil's  paper  was  then  given -a  positive  grade  based  upon  the 
values  thus  assigned  to  the  various  exercises.  For  reasons  to  be 
explained  later  the  negative  scores  were  not  weighted.2  Also 
as  there  is  no  upper  limit  to  the  number  of  incorrect  and  unnec- 
essary elements  that  can  be  given,  it  is  impossible  to  express  the 
negative  scores  in  per  cents.  Hence  we  have  used  as  the  pupil's 
negative  score  the  total  number  of  incorrect  and  unnecessary 
elements  given  in  the  entire  test. 

Means  of  Securing  Uniformity  of  Scoring. — All  papers  were 
scored  by  the  author.  In  each  test  the  first  exercise  was  scored 
for  the  entire  school,  then  the  second,  and  so  on,  until  all  the 
exercises  had  been  marked  for  the  given  school.  Before  begin- 
ning to  mark  the  papers,  each  exercise  was  carefully  solved  and 
the  number  of  necessary  steps  noted.  This  number  was  used 
as  a  basis  for  expressing  the  pupils'  positive  scores  for  that 
exercise  in  per  cents.  Some  of  the  exercises  admit  of  more  than 
one  solution.  In  such  cases  a  copy  of  each  new  solution  found 
among  the  papers  was  kept  for  reference,  and  a  pupil's  positive 
score  was  based  on  the  number  of  necessary  steps  in  the  method 
which  he  had  chosen.  If  any  answer  gave  peculiar  difficulty  a 
memorandum  of  how  it  was  scored  was  kept  as  a  guide  in  all 
similar  cases. 

There  will,  no  doubt,  be  a  difference  of  opinion  as  to  the  number 
of  steps  necessary  for  a  complete  solution  of  some  of  the  exercises, 
but  the  details  given  in  the  following  pages  are  the  results  of 
experience  gained  in  scoring  about  one  thousand  papers  from 
the  preliminary  tests,  and  any  one  caring  to  compare  his  results 
with  those  of  this  investigation  should  carefully  follow  this 
method  of  scoring.5* 

Scoring  Each  Test. — We  shall  now  describe  in  detail  the 
scoring  of  each  test. 

Test  A. — The  figure  necessary  for  the  correct  solution  of  each 
exercise  of  Test  A  is  given  below.  With  each  figure  is  a  state- 
ment of  the  points  necessary  for  a  complete  drawing.4 

1  Pages  43-49- 

2  Page  44. 

8  For  a  condensed  statement  of  directions  for  scoring  papers  see  page  97. 
4  For  the  exercises  see  page  1 1 . 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  29 

I 


Circle  0 I 

OA  a  radius I 

~r>  (  &  radius  1 

\  perpendicular  to  OA  J 

(  tangent  to  0        "} 
AB  <  cutting  OB  at  B  > 3 

I  cutting  OA  at  A  ) 


f  tangent  to 
\  through  J5 


(  tangent  to  0 
\  through  ^4 


1 

/ 


Total  number  of  points  ......................  .  .  .  .  .  1  1 


II 


AB  a  straight  line I 

CD  a  straight  line I 

fa  straight  line               ^| 
EC-Lpposite  AB  from  CDl 3 

Imeeting  CD  on  AB     J 
Zi  =   /2.  ,    i 


Total  number  of 'points 6 


INVESTIGATION   OF   CERTAIN   ABILITIES 
III 


Triangle  ABC  ..................................    I 

ADf  through.4  I  2 

I  to  mid-point  of  BC  j 
BE  similar  to  AD  ..............................   2 

CF  similar  to  AD  ...............................   2 

a  straight  line    \ 

outside  of  ABC  I  ' 


\ 

J  ' 


perpendicular  to  PQ 

BM  similar  to  AK  ..............................  2 

CN  similar  to  AK  ..............................  2 

OL  similar  to  AK  ...............................  2 

Total  number  of  points  ..........................  17 


IV 


Circle  0 I 

43Cifrian?le    \  2 
\  inscribed  J  ' 

A  C  produced  to  K I 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  3! 


\ 
J 


through  C 
making    Z  I  =   Z  2 
through  C  \ 

making  Z  3  =   Z  4  J  ' 
£/rr  through  E  |  2 

I  thro  ugh  FJ 

Total  number  of  points 10 

V 

F 


Circle  0 


AD  and  BC  produced  to  meet  ....................    i 

DC  and  AB  produced  to  meet  ....................    i 

RKC  through  E 

I  making  /I  =  /2 
LF  similar  to  EK  ...............................   2 

Total  number  of  points  ..........................   9 

As  an  illustration  of  this  method  of  scoring,  suppose  a  pupil 
drew  the  following  figure  for  exercise  II. 


32  INVESTIGATION   OF   CERTAIN   ABILITIES 

The  correct  points  in  this  drawing  are  as  follows: 

AB  a  straight  line I 

CD  a  straight  line I 

fa  straight  line               "1 
EC!  opposite  AB  from  DC  }• 3 

I  meeting  DC  on  AB      J 

Total  number  of  correct  points 5 

Incorrect  drawing: 

Zi  =   Z3 I 

The  total  number  of  correct  points  should  be  6.  The  pupil  has 
five  of  these  correct  and  he  has  one  error.  Hence  his  positive 
score  is  83,  and  his  negative  score  is  i. 

Certain  peculiarities  should  be  noted.  If  the  pupil  omitted 
the  letters  A  and  B  from  his  figure  for  exercise  I  nothing  was 
deducted  from  his  positive  score.  If,  however,  these  letters 
were  incorrectly  used  they  were  counted  in  determining  the 
negative  score.  If  in  the  figure  for  exercise  III  the  medians  were 
not  produced  to  the  mid-points  of  the  sides  of  the  triangle  but 
would  pass  through  these  points  if  produced,  they  were  counted 
as  correct.  Also  if  the  figure  was  drawn  so  that  two  of  the 
perpendiculars  to  the  line  PQ  coincided  full  credit  was  given  for 
these  coincident  perpendiculars.  In  exercise  IV  some  pupils 
drew  the  bisectors  of  the  exterior  and  interior  angles  at  each  of 
the  three  vertices.  In  such  cases  the  drawings  at  two  of  the 
vertices  were  counted  as  unnecessary.  The  pupils  sometimes 
produced  the  opposite  sides  of  the  inscribed  quadrilateral  of 
exercise  V  in  the  direction  in  which  they  would  not  meet.  This 
was  counted  as  an  unnecessary  drawing. 

Test  B. — The  answer  to  each  exercise  of  Test  B  and  the  number 
of  necessary  statements  in  each  answer  are  given  below.1 

I 
Given : 

Circles  0  and  0' 2 

B  a  point  of  intersection  of  0  and  0' I 

OO'  the  line  of  centers I 

1  See  pages  12-14  for  exercises  and  figures. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  33 

fa  common  secant  "1 
CB  -j  through  B  > 3 

L  parallel  to  00'        J 

Ta  common  secant  ^| 
EF  \  through  B  1 3 

I  not  parallel  to  00'  J 

Total  number  of  points 10 

To  prove: 

CB  >  EF i 

II 

Given : 

Circles  0  and  0'  internally  tangent 3 

BC  the  common  tangent I 

f  chord  of  circle  0      "1 
EG  -I  tangent  to  circle  0*r 5 

L  parallel  to  A  C          J 

r  chord  of  circle  0      ^ 
DF  J  tangent  to  circle  Of  j- 3 

Lnot  parallel  to  CB  J 

Total  number  of  points 10 

To  prove: 

EG  >  DF i 

III 

Given : 

Triangle  ABC I 

Angle  BCA I 

CD  bisects  angle  BCA i 

AB  the  side  opposite  angle  BCA i 

BE  perpendicular  to  CD i 

AD  perpendicular  to  CD i 

F  the  mid-point  of  AB i 

Lines  FE  and  FD ^ 2 

Total  number  of  points 9 

To  prove: 

FE  =  FD i 

FE  or  FD  =  \(AC  -  CB) .  : i 

Total  number  of  points 2 


34  INVESTIGATION   OF   CERTAIN   ABILITIES 

IV 
Given : 

Triangle  ABC I 

Angle  ABC i 

Median  CD I 

AB  the  side  opposite  angle  A  CB I 

CD  =  \AB, i 

CD  >  \AB,  or i 

CD  <  \AB i 

Total  number  of  points 7 

To  prove: 

Angle  A  CB  is  a  right  angle, i 

Angle  A  CB  is  an  acute  angle,  or i 

Angle  A  CB  is  an  obtuse  angle i 

Total  number  of  points 3 

The  unweighted  positive  score  for  each  exercise  was  obtained 
by  grading  the  hypothesis  and  conclusion  each  on  a  scale  of  100 
and  then  taking  the  average  of  the  two  grades.  A  statement 
was  considered  as  correct  only  when  it  was  given  in  terms  of  the 
figure.  Such  general  statements  as,  "Given  two  intersecting 
circles"  were  counted  in  determining  neither  the  positive  nor 
the  negative  score.  Pupils  frequently  include  a  part  of  the 
hypothesis  in  the  form  of  a  modifying  phrase  or  clause  in  the 
statement  of  the  conclusion.  Generally  this  should  not  be  per- 
mitted as  it  causes  the  pupil  to  lose  sight  of  the  parts  of  the 
hypothesis  given  in  the  conclusion.  For  this  reason  any  part 
of  the  hypothesis  given  in  the  conclusion  was  not  counted  as 
correct,1  nor  was  it  included  in  the  count  for  the  negative  score. 
An  illustration  will  make  the  method  of  scoring  clear.  Suppose 
a  pupil  answered  exercise  III  as  follows:2 
Given:  The  triangle  ABC  with  the  bisector  of  angle  ACS, 

BF  and  AD  are  perpendicular  to  CD. 
To  prove :  The  lines  FE  and  ED  connecting  the  mid-point  of  AB 

with  the  feet  of  the  perpendiculars  BF  and  AD  are  equal  and 

either  equals  \(AC  -  AB): 
Points  correctly  and  specifically  stated  in  hypothesis : 

1  An  exception  was  made  to  this  rule  in  exercise  IV.     See  pages  35~36. 
1  Page  13. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  35 

ABC  is  a  triangle I 

Angle  ACB I 

BF  is  perpendicular  to  CD I 

AD  is  perpendicular  to  CD I 

Number  of  correct  points  in  hypothesis 4 

Per  cent,  of  points  correct  in  the  hypothesis 44 

Points  correctly  and  specifically  stated  in  the  conclusion. 

FE  equals  ED i 

Per  cent,  of  points  correct  in  conclusion 50 

Average  score  for  hypothesis  and  conclusion 47 

Incorrect  statements  in  the  answer: 

FE  equals  \(AC  -  AB) i 

ED  equals  \(AC  -  AB) i 

Total  number  of  incorrect  points 2 

Hence  the  positive  score  is  47  per  cent,  and  the  negative 
score  is  2.  The  statement  concerning  the  bisector  of  angle  ACB 
is  not  specific  and  is  therefore  not  counted.  The  statements 
concerning  the  lines  FE  and  ED  and  the  mid-point  of  AB  are 
not  counted  because  they  are  involved  in  the  statement  of  the 
conclusion. 

Certain  special  cases  should  be  noted.  Care  was  taken  not 
to  count  the  same  lack  of  specific  statement  twice.  For  example, 
if  in  exercise  IV  a  pupil  made  the  following  statement,  "Given 
the  bisector  of  angle  ACB,  and  AD  perpendicular  to  the  bisector 
of  ACB,"  there  are  seemingly  two  points  in  which  he  failed  to 
be  specific.  He  did  not  name  the  bisector  of  angle  A  CB  nor  did 
he  name  the  line  to  which  AD  is  perpendicular.  If,  however, 
he  had  named  the  bisector  of  angle  ACB  the  second  statement 
would  have  been  specific.  Hence  he  should  receive  credit  for 
the  second  statement.  Freedom  of  expression  was  permitted 
as  long  as  the  pupil  made  a  specific  statement  of  each  point  in 
the  hypothesis  and  conclusion.  Thus  in  exercise  II  he  was  not 
required  to  say  that  DF  is  not  parallel  to  CB.  Any  statement 
clearly  distinguishing  DF  from  GE  was  accepted.  In  exercise 
IV  there  are  three  conclusions  each  dependent  upon  a  separate 
part  of  the  hypothesis.  Pupils  experience  considerable  difficulty 
in  getting  a  clear  statement  of  this  case  if  they  are  required  to 
separate  completely  the  hypothesis  and  conclusion.  Hence  for  a 
statement  such  as  the  following: 


36  INVESTIGATION    OF   CERTAIN   ABILITIES 

To  prove  that 

1  Angle  ACB  is  a  right  angle  if  CD  =  \AB, 

2  Angle  ACB  is  an  acute  angle  if  CD  >  \AB, 

3  Angle  ACB  is  an  obtuse  angle  if  CD  <  \AB, 

credit  was  given  for  a  perfect  statement  of  the  conclusion  and 
for  three  points  in  the  hypothesis.  It  may  seem  that  the  con- 
clusion of  exercise  III  should  count  as  three  points;  namely, 
FE  =  FD,  FE  =  \(AC  -  CB),  and  FD  =  \(AC  -  CB).  But 
the  first  and  either  of  the  other  two  statements  are  equivalent 
to  the  remaining  statement  and  since  the  pupils  seemed  to  have 
this  clearly  in  mind  the  conclusion  was  counted  as  two  points. 

Test  C. — In  Test  C1  the  pupil  was  free  to  give  any  facts  which 
he  could  recall  concerning  the  given  figure.  The  pupils'  state- 
ments varied  so  greatly  that  it  is  impossible  to  give  model 
answers  for  the  various  exercises  of  this  test.  We  do  not  know 
how  many  facts  about  any  one  of  these  figures  a  pupil  should 
be  able  to  give.  Therefore,  we  have  no  exact  basis  for  computing 
the  positive  scores  in  per  cents  and  it  is  necessary  to  select 
one  arbitrarily.  The  smallest  number  of  facts  stated  correctly 
by  any  one  of  the  highest  ten  per  cent,  of  all  pupils  taking  the 
test  does  not  seem  too  high  a  standard  to  set  for  a  perfect  answer. 
However,  it  is  seldom  possible  to  make  this  exact  division  of  the 
pupils.  For  example,  suppose  the  following  condition  to  exist: 
If  we  take  the  highest  group  of  pupils  such  that  the  smallest 
number  of  facts  given  by  any  one  of  them  is  nine  we  include 
less  than  ten  per  cent,  of  all  the  pupils;  but  if  we  increase  this 
group  until  the  smallest  number  given  by  any  one  is  eight  we 
include  more  than  ten  per  cent.  It  is  then  impossible  to  select 
the  number  of  facts  given  by  exactly  the  highest  ten  per  cent. 
In  all  such  cases  the  larger  number  of  facts  was  selected.  That 
is,  the  basis  for  computing  the  positive  score  in  per  cents  was 
the  smallest  number  of  facts  given  by  any  one  of  the  highest 
group  of  pupils,  this  group  not  to  exceed  ten  per  cent,  of  all 
pupils  taking  the  test  but  to  be  as  nearly  ten  per  cent,  as  possible. 
The  number  of  correct  facts  given  by  each  pupil  has  been  care- 
fully noted  and  the  data  is  given  in  Table  IV.  In  order  to 
indicate  how  nearly  a  constant  condition  has  been  obtained  this 
table  has  been  arranged  in  a  cumulative  way.  Thus,  the  first 

1  Pages  14-15. 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY 


37 


line  gives  the  data  for  school  VII,  the  second  line  for  schools 
VII  and  VIII,  etc.,  the  last  line  giving  the  combined  data  for 
all  the  schools  in  which  the  test  was  given.  In  column  a  under  I 
is  the  least  number  of  facts  for  exercise  I  given  by  any  one  of 
the  highest  group  of  pupils  which  does  not  exceed  ten  per  cent, 
of  all  the  pupils  taking  the  test  but  is  as  nearly  ten  per  cent,  as 
possible.  In  column  b  under  I  is  the  per  cent,  of  all  the  pupils 
who  gave  that  number  of  facts  correctly.  Thus,  the  seventh 
line  indicates  that,  for  exercise  I,  eight  or  more  facts  were  given 
correctly  by  each  pupil  of  the  highest  9.8  per  cent,  of  those 
taking  the  test,  and  8  is  the  smallest  number  of  facts  that  can 
be  taken  without  including  more  than  ten  per  cent,  of  the  pupils. 
An  examination  shows  that  enough  pupils  have  been  tested  to 
give  fairly  constant  results  for  all  the  exercises  with  the  possible 
exception  of  exercise  II.  Therefore  the  numbers  of  correct 
facts  accepted  as  perfect  positive  scores  for  the  exercises  of 
Test  C  were  8,  30,  7  and  18  respectively. 

TABLE  IV. — Least  number  of  correct  facts  given  for  each  exercise  of  Test  C  by  any  one 
of  the  highest  group  which  does  not  exceed1  ten  per  cent,  of  all  pupils  taking  the 
test  but  is  as  nearly  ten  per  cent,  as  possible. 


Exercise 

I 

II 

III 

IV 

Number 
of 
Pupils 

School 

a               b 

a             b 

a              b 

a              b 

VII 

II 

IO.O 

32 

6.7 

9 

7-7 

21 

IO.O 

30 

VIII 

9 

6.6 

34 

9.1 

8 

8.0 

19 

9.0 

578 

IX 

9 

6.0 

33 

9.4 

8 

7-5 

19 

8.4 

651 

X 

9 

5-9 

33 

8.9 

8 

7-3 

19 

8.2 

682 

XI 

9 

5-3 

31 

10.0  + 

7 

10.  0  + 

19 

8.2 

710 

XII 

8 

IO.O 

31 

IO.O 

7 

9-3 

18 

IO.O 

795 

XIII 

8 

9.8 

31 

9.6 

7 

9.0 

18 

IO.O  + 

844 

XXVIII 

8 

9.6 

31 

9.2 

7 

8.7 

18 

9.8 

882 

XXXII 

8 

9-5 

31 

9.0 

7 

8.7 

18 

9.2 

908 

XXXIV 

8 

10.0  + 

30 

9-7 

7 

9.4 

18 

9-4 

993 

XXXVII 

8 

9.8 

30 

9.6 

7 

9.4 

18 

IO.O 

1019 

XXXVIII 

8 

9.8 

30 

9-5 

7 

9-4 

18 

IO.O 

1047 

In  Test  C  a  statement  was  counted  as  correct  only  when  it 
gave  accurately  and  concretely  some  relation  between  parts  of 
the  figure  or  when  it  gave  the  value  of  some  magnitude  correctly. 
The  mere  naming  of  the  parts  of  a  figure  (e.  g.,  AB  is  a  chord) 
was  counted  in  determining  neither  the  positive  nor  the  negative 
grade.  General  statements,  such  as  "The  sum  of  two  sides  of  a 

1  In  this  table  10.0+  indicates  that  slightly  more  than  ten  per  cent,  of  the 
pupils  gave  the  corresponding  number  of  correct  facts. 


38  INVESTIGATION   OF   CERTAIN   ABILITIES 

triangle  is  greater  than  the  third  side,"  are  useless  in  the  demon- 
stration of  a  proposition  unless  the  pupil  can  show  how  they 
apply  to  a  given  figure.  Hence  only  facts  stated  in  terms  of  the 
figure  were  counted  as  correct.  However,  statements  given  in 
general  terms  were  counted  in  determining  the  negative  grade 
only  when  they  were  incorrectly  given.  Pupils  sometimes  made 
additional  drawings  and  then  gave  facts  concerning  the  new 
figure.  Such  facts  were  eliminated  for  two  reasons.  Many 
pupils  who  could  have  given  such  facts  did  not  because  the  test 
did  not  call  for  them.  Hence  their  results  would  not  have  been 
comparable  with  the  results  of  those  who  did  give  such  state- 
ments. Second,  there  is  no  limit  to  the  number  of  such  facts 
since  there  is  no  end  to  the  drawings  which  could  be  added.  If  a 
pupil  made  a  continued  statement  such  asa  =  b  =  c  =  dor 
x  >  y  >  2  credit  was  given  for  the  full  number  of  facts  involved. 
On  the  other  hand  care  was  taken  not  to  give  credit  twice  for  a 
fact  which  was  repeated  in  the  same  or  slightly  different  form. 
Thus  a  +  b  =  c  and  a  =  c  —  b  express  the  same  relation. 
Likewise  the  statements  a  >  b  or  c,  and  c  <  a  or  b  repeat  the 
relation  between  a  and  c. 

Test  D. — In  this  test  the  pupil  was  asked  to  produce  the  proof 
for  the  exercises.  As  each  exercise  admitted  of  two  or  more 
proofs  and  the  pupil  was  free  to  select  any  proof  he  desired,  it  is 
necessary  to  consider  the  different  proofs  possible  for  each 
exercise.  The  various  proofs  found  in  the  papers  follow1 : 

I 

(a)  AB  =  AD i 

Zi  =  Z4 i 

Z2  =   /5 i 

AADE  =  AABC i 

Number  of  necessary  steps 4 

(b)  ED  is  parallel  to  BC I 

Z2  =  Z5 i 

Zi  =  Z4 i 

AB  =  AD i 

&ADE  =  AABC i 

Number  of  necessary  steps 5 

1  For  the  exercises  see  pages  15-17. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  39 

(c)    Make  DA  coincide  with  AB I 

Zi  =   Z4 •  •  l 

AE  takes  the  direction  of  AC I 

Z5  =   Z2 I 

DE  takes  the  direction  of  BC I 

Point  E  falls  on  point  C I 

AADE  =  AABC i 

Number  of  necessary  steps 7 


II 

(a)    DO  +  OP  >  DP I 

DO  =  AO i 

AO  +  OP  >  DP i 

AO  +  OP  =  AP1 i 

AP  >  DP i 

Number  of  necessary  steps 5 

(6)    Draw  AD I 

AO  =  DO i 

Z9  =   Z8 i 

Z9  +  Z4  >  Z8 I 

AP  >  DP i 

Number  of  necessary  steps 5 

III 


Z2  = 


Z4  =   Z7  ....... 

Z  4  is  measured  by  \AE  . 
Z7  is  measured  by  \AD 
ArcAE  =  arc  AD.  . 


Number  of  necessary  steps 6 

1  Sometimes  when  this  statement  was  omitted  it  was  clear  that  the  pupil  had 
it  in  mind.     In  such  cases  credit  was  given  for  the  step. 


40  INVESTIGATION   OF   CERTAIN   ABILITIES 

(b)  Z8  =   Z3 i 

Zio  =  zn I 

Z7  =   /4-  i 

Z  7  is  measured  by  \AD I 

Z  4  is  measured  by  \AE I 

Arc  AE  =  arc  AD I 

Number  of  necessary  steps 6 

(c)  Zi  +  Z2  +  Z7  =  180° 

Zi  +  Z4  +  Z9  =  1 80°. 

Zi  +  Z2  +  Z7  =   zi  +  Z4+  /9-- 

Zi  =   Zi1 

Z2  =   Z9 

Z7  =   /4 

Z  7  is  measured  by  \AD 

Z  4  is  measured  by  \AE I 

Arc  AE  =  arc  AD i 

Number  of  necessary  steps 9 

(d)  Z2=  Z4+  /5+  /6 i 

Z9=  Z5+  /6+  /7 I 

Z2  =  Z9 i 


Z4  =  £1 i 

Z  4  is  measured  by  \AE I 

Z  7  is  measured  by  \AD I 

Arc  AE  =  arc  AD i 

Number  of  necessary  steps 9 

(e)    CD  is  perpendicular  to  AB 

BE  is  perpendicular  to  AC 

Z4  =   2.7 

Z  4  is  measured  by  \AE 

Z  7  is  measured  by  \AD , 

Arc  AE  =  arc  AD 

Number  of  necessary  steps 6 

1  If  this  statement  was  omitted  but  clearly  in  the  mind  of  the  pupil  credit  was 
given  for  it. 


FUNDAMENTAL   TO   THE    STUDY  OF   GEOMETRY  4! 


° 

° 


(/)     Z3  =90 

Z4  +  Zn  =  90 

Z8  =  90° 

Z7  +  Zio  =  90° 

Z4  +  Zn  =   Zio  +  Z7 

Zn  =   Zio  ......... 

Z4  =   Z7- 

Z  4  is  measured  by  \AE 
Z  7  is  measured  by 
ArcAE  =  arc  AD. 


Number  of  necessary  steps 10 

(g)     Z8  =   Z3 • 

Z  8  is  measured  by  %(BC  +  AE) 

Z  3  is  measured  by  J(BC  +  AD) 

BC  +  AE  =  BC  +  AD 

AE  =  AD 

Number  of  necessary  steps 5 

(h)     Z 7  is  a  complement  of  Zio 

Z  4  is  a  complement  of  Z  1 1 

Zio  =   zn 

Z4  =   Z7-  • 

Z  4  is  measured  by  \AE 

Z  7  is  measured  by  \AD 

AE  =  AD i 

Number  of  necessary  steps 7 

The  number  of  necessary  steps  in  the  proof  of  an  exercise  was 
used  as  the  basis  for  computing  the  positive  score.  The  pupil 
was  free  to  select  any  method  of  proof  he  desired  and  the  number 
of  necessary  steps  varied  with  his  choice.  In  each  case  the  num- 
ber of  necessary  steps  in  the  proof  chosen  was  taken  as  the  basis. 
The  numbers  used  for  each  method  found  in  the  papers  are 
given  in  connection  with  the  proofs  on  pages  38-41.  These 
numbers  may  be  slightly  varied  depending  upon  the  number  of 
statements  which  are  implied  but  not  expressed.  The  selection 
of  the  above  numbers  was  based  on  experience  gained  in  grading  a 
large  number  of  papers,  and  further  experience  seems  to  justify 

4 


42  INVESTIGATION   OF   CERTAIN   ABILITIES 

this  selection.  If  a  proof  was  incomplete  the  pupil  was  given 
credit  for  the  number  of  correct  facts  given.  If  he  had  not 
carried  the  proof  far  enough  to  show  what  method  he  had  in 
mind  the  scorer  completed  the  proof  with  the  least  possible 
number  of  steps  and  used  that  number  as  a  base  for  computing 
the  positive  score.  If  a  statement  was  given  out  of  its  logical 
order  it  was  counted  incorrect.  If,  however,  a  conclusion  was 
followed  by  the  facts  leading  up  to  it  and  the  relation  was  indi- 
cated by  "since,"  "for,"  or  some  other  like  expression,  full 
credit  was  given.  For  example,  the  following  statements  would 
be  counted  correct, 

AABC  =  ADEF, 
since 

AB  =  DE, 


BC  =  EF, 


and 


The  pupil  was  asked  to  omit  the  authorities  and  reasons  for  the 
steps  in  his  proof.  If  such  authorities  were  given  they  were  not 
used  in  determining  either  the  positive  or  the  negative  score. 
The  pupil  often  made  a  correct  statement  and  later  repeated  it, 
apparently  to  call  attention  to  it.  Such  repetitions  were  dis- 
regarded. 

Test  E. — In  this  test1  the  positive  score  was  either  one  hundred 
or  zero.  If  a  drawing  made  a  proof  possible  it  was  one  hundred. 
For  any  other  drawing  it  was  zero.  The  following  drawings 
making  a  proof  possible  were  given  by  the  pupils. 


c- 

A- 


0 


F 


1  For  the  exercises  see  pages  17-19. 


F 


F 


•D 


7-B 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 

B 

?>E 

D 


43 


F 
II 


III 


IV 


D 


Weighting  the  Exercises. — As  the  exercises  of  any  one  of  the 
tests  differ  in  degree  of  difficulty  with  respect  to  both  the  positive 
and  negative  scores,  different  values  should  be  assigned  to  each 
exercise.  Thus,  a  pupil  should  receive  more  credit  for  a  correct 
drawing  for  exercise  V  of  Test  A  than  for  a  correct  drawing 


44  INVESTIGATION   OF   CERTAIN   ABILITIES 

for  exercise  I  of  the  same  test.  Certain  difficulties  present  them- 
selves which  we  must  now  consider. 

As  the  positive  and  negative  scores  for  each  exercise  cannot 
be  combined,  a  single  weight  cannot  be  assigned  for  the  two  ele- 
ments. The  incorrect  and  unnecessary  steps  introduced  into 
the  solution  of  a  given  exercise  varied  greatly.  One  error  oc- 
curred more  frequently  than  another,  but  the  same  error  seldom 
occurred  in  a  large  number  of  papers.  Therefore  we  do  not  have 
enough  material  to  weight  each  error  separately.  Moreover 
such  a  procedure  would  involve  an  undue  amount  of  labor  as 
there  was  no  limit  to  the  number  of  different  errors  which  could 
be  introduced.  Also,  we  have  no  means  (such  as  the  per  cents 
of  total  number  of  possible  errors)  of  comparing  the  negative 
values  of  the  exercises  of  a  test.1  Therefore  the  exercises  have 
not  been  weighted  according  to  the  negative  scores.  The  total 
number  of  errors  made  by  a  pupil  in  answering  all  the  exercises 
of  a  test  was  taken  as  the  final  negative  score. 

The  correct  solution  of  any  of  the  exercises,  excepting  those 
of  Test  E,  involves  several  steps  which  differ  in  degree  of  diffi- 
culty. If  we  try  to  weight  these  steps  separately  on  the  basis 
of  the  per  cent,  of  pupils  giving  them  correctly,  complications 
arise  from  the  facts  that  in  Tests  C  and  D  the  statements  re- 
quired for  a  correct  solution  were  not  always  the  same  for  a  given 
exercise.  Further  in  Test  D  the  steps  are  so  related  that  it  is 
impossible  to  say  that  a  difficulty  lies  wholly  within  any  one  of 
them.  Hence  this  basis  for  weighting  seems  impracticable. 
There  is  also  a  possibility  of  weighting  each  statement  according 
to  its  relative  geometrical  value  but  we  do  not  know  how  to 
determine  this  value.  Hence  it  was  decided  to  weight  each 
exercise  as  a  whole  according  to  the  positive  scores. 

The  question  of  securing  data  under  the  same  conditions  for 
each  of  a  set  of  questions  also  involved  a  difficulty.  An  exercise 
occurring  in  a  series  of  exercises  has  two  types  of  difficulty. 
One  is  its  intrinsic  difficulty  due  to  its  own  peculiarities;  the 
other  may  be  called  its  place  difficulty  due  to  its  position  in  the 
series.  Fatigue,  suggestion  from  a  preceding  exercise,  distraction 
caused  by  a  preceding  difficulty,  and  encouragement  due  to 
preceding  success  are  some  of  the  factors  which  influence  this 
latter  type  of  difficulty.  In  order  to  eliminate  this  place  difficulty 

1  Page  28. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  45 

investigators  sometimes  give  a  series  of  questions  in  one  order 
and  then  reverse  the  order  and  average  the  grades  of  each 
question  for  the  two  trials.  This  procedure  may  eliminate  one 
or  more  of  the  influences  due  to  position  but  the  reversed  order 
brings  the  pupils  to  a  given  question  through  a  new  succession 
of  questions  which  introduces  new  elements.  Hence  we  do  not 
know  that  this  method  is  equivalent  to  giving  each  exercise 
under  the  same  conditions.  For  our  purpose  it  is  not  even 
desirable  that  the  place  difficulty  be  eliminated.  The  exercises 
must  be  given  in  some  order.  This  order  will  present  its  own 
peculiarities  and  the  value  assigned  to  each  exercise  should  be 
dependent  upon  both  its  intrinsic  and  place  difficulties.  Hence 
the  positive  values  assigned  to  the  exercises  of  each  test  are 
based  upon  the  results  obtained  by  giving  the  tests  in  their  final 
order  without  any  reversal  of  that  order. 

We  do  not  have  sufficient  data  to  locate  a  zero-point  in  the 
way  it  has  been  located  in  certain  scales.1  Due  to  extensive 
elimination  during  the  earlier  school  years,  the  pupils  tested 
constituted  a  specially  selected  group.  Hence,  the  probability 
that  any  high  school  pupil  will  have  zero  ability  in  any  phase  of 
the  work  with  which  this  study  is  concerned  is  very  slight. 
Therefore,  since  no  considerable  number  of  pupils  made  a  zero 
score  in  a  given  test,  it  is  impossible  to  determine  the  exact 
point  at  which  total  inability  to  do  any  part  of  that  test  begins. 


VERY  POOR  A          M  B  VERY  GOOD 

FIG.  i.     Normal  surface  of  frequency. 

However,  we  may  safely  assume  that  distribution  based  on  any 
of  the  abilities  in  question  follows  the  same  law  as  a  distribution 
based  on  any  other  human  trait.  That  is,  if  there  were  no  elimi- 
nations distributions  according  to  any  one  of  these  abilities  would 
result  in  the  normal  frequency  curve  as  shown  in  Fig.  I.  The 
curve,  when  extended  indefinitely  in  either  direction  from  the 
median  MN,  continues  to  approach  the  line  CD.  The  direction 
from  left  to  right  will  be  considered  positive,  and  from  right  to 

1  Trabue,  "Completion — Test  Language  Scales,"  p.  52. 


46 


INVESTIGATION   OF   CERTAIN   ABILITIES 


left  negative.  If  AP  is  drawn  so  that  A  MNP  is  one  fourth  of 
the  entire  surface  under  the  curve;  that  is,  so  that  twenty-five 
per  cent,  of  the  cases  fall  between  M  and  A ,  then  A  M  is  known 
as  the  P.E.1  For  practical  purposes  the  curve  meets  the  line 
CD  at  about  4.6  P.E.  above  and  below  M.  If  a  question  can 
be  answered  by  all  pupils  above  —  4.6  P.E.  the  ability  required 
is  very  slight.  Therefore,  the  point  —  4.6  P.E.  has  been  arbi- 
trarily assumed  as  the  zero-point.  The  weighted  values  assigned 
to  the  exercises  of  each  test  are  proportional  to  the  distances  in 
P.E.  above  this  arbitrary  zero-point  and  they  are  such  that  their 
sum  is  one  hundred.  We  shall  now  consider  the  weighting  of 
the  exercises  of  each  test. 

TABLE  V. — Average  positive  scores  of  each  exercise  of  Test  A. 


Schools 

Exercises 

Number  of 
Pupils 

I 

II 

III 

IV 

V 

XXIII 

77 

76 

65 

52 

30 

88 

XXV 

79 

75 

65 

50 

30 

158 

XXXIII 

80 

76 

62 

49 

29 

173 

XXXV 

81 

77 

64 

50 

29 

196 

XXXVI 

82 

77 

64 

48 

29 

222 

XXXIX 

81 

75 

64 

49 

29 

239 

XL 

82 

75 

65 

Si 

30 

286 

XLI 

84 

74 

68 

50 

31 

334 

XLII 

86 

76 

79 

50 

32 

385 

XLIII 

86 

77 

72 

52 

34 

438 

XLIV 

87 

78 

73 

55 

36 

490 

XLV 

88 

79 

75 

55 

37 

54i 

L 

88 

79 

75 

55 

37 

555 

LI 

88 

79 

74 

55 

36 

569 

LII 

88 

79 

73 

55 

37 

617 

LIII 

87 

79 

73 

55 

36 

688 

LIV 

87 

80 

73 

55 

36 

710 

LVI 

87 

79 

72 

54 

35 

773 

LIX 

87 

79 

72 

54 

36 

802 

LX 

87 

79 

72 

54 

35 

831 

LXI 

87 

79 

72 

54 

35 

856 

LXIII 

88 

79 

7i 

54 

34 

944 

Test  A  was  given  to  1,094  pupils  but  as  school  LXI  I  gave  the 
test  to  pupils  who  had  completed  all  of  plane  geometry  the  data 
from  this  school  were  not  used  in  weighting  the  exercises.  Table 
V  gives  the  average  positive  scores  for  each  exercise  of  the  test. 
If  the  values  assigned  to  the  exercises  are  to  be  reliable  the 
number  of  pupils  tested  must  be  sufficient  to  eliminate  chance 

1  Trabue,  "Completion-Test  Language  Scales,"  pp.  30-35. 


FUNDAMENTAL   TO   THE   STUDY   OF  GEOMETRY 


47 


variations.  In  order  to  indicate  how  nearly  this  condition  has 
been  realized  this  table  is  arranged  in  a  cumulative  way.  A 
study  of  the  table  shows  that  a  fairly  constant  condition  has 
been  obtained,  and  perhaps  the  addition  of  more  schools  would 
not  change  the  results  materially. 

TABLE  VI. — Positive  values  assigned  to  each  exercise  of  Test  A. 


Exercise 

Average  Score 

Difference 
Between  Score 
and  50% 

Distance  in 
P.E.  from 
Median 

Distance  Above 
Zero-point 

Value 
Assigned 

I 

87.5 

-37-5 

—  1.706 

2.894 

15 

II 

79.0 

—  2Q.O 

—  1.196 

3404 

17 

III 

71.0 

—  21.  0 

—  0.820 

3.780 

19 

IV 

53-5 

-    3-5 

—  0.130 

4.470 

23 

V 

34-1 

+  15-9 

+  0.608 

5.208 

26 

Table  VI  gives  the  values  assigned  to  each  exercise  of  Test  A 
and  it  indicates  how  this  value  was  obtained.  If  we  use  the 
scale  from  o  to  100  and  have  a  normal  distribution  of  pupils, 
the  median  pupil  falls  at  50.  The  first  number  in  the  column 
headed  "Average  score"  shows  that,  when  judged  by  exercise  I 
alone,  this  median  pupil  will  make  a  score  of  87.5.  That  is 

TABLE  VII. — Average  positive  scores  for  each  exercise  of  Test  B. 


Schools 

Exercises 

Number  of 
Pupils 

I 

II 

III 

IV 

XIV 

67 

55 

47 

32 

19 

XV 

67 

64 

58 

44 

34 

XVI 

79 

75 

66 

48 

85 

XVII 

75 

72 

62 

45 

121 

XVIII 

82 

78 

70 

50 

26? 

XIX 

83 

78 

76 

52 

344 

XX 

83 

77 

73 

53 

521 

XXI 

81 

76 

72 

52 

593 

XXV 

81 

75 

70 

Si 

659 

XXX 

80 

74 

68 

49 

713 

XXXI 

80 

73 

66 

48 

766 

XLVII 

80 

75 

66 

47 

802 

XL  VI  II 

80 

74 

67 

48 

849 

XLIX 

81 

75 

67 

48 

935 

LVII 

81 

75 

67 

47 

975 

LVIII 

81 

74 

67 

47 

1025 

exercise  I  is  87.3  —  50  or  37.3  too  easy  for  the  median  pupil. 
Converting  this  into  its  P.E.  value1  we  get   —  1.706  which  is 

JFor  this  purpose  Table  XIII  of  Trabue's  Completion-Test  Language  Scale 
has  been  used. 


48 


INVESTIGATION   OF   CERTAIN   ABILITIES 


2.894  P-E.  above  the  assumed  zero-point.  The  distances  of  the 
other  exercises  above  the  zero-point  have  been  found  in  the 
same  way.  The  values  given  in  the  last  column  of  Table  VI 
are  proportional  to  these  distances  and  they  are  such  that  their 
sum  is  100. 

TABLE  VIII. — Positive  values  assigned  to  each  exercise  of  Test  B. 


Exercise 

Average  Score 

Difference 
Between  Score 
and  50  •/> 

Distance  in 
P.E.  from 
Median 

Distance  Above 
Zero-point 

Value 
Assigned 

I 

80.7 

-  30-7 

-  1.286 

3.314 

21 

II 

74-4 

-  24.4 

-  0.972 

3.628 

23 

III 

66.5 

-16.5 

—  0.632 

3-968 

26 

IV 

46.7 

+    3-3 

+  0.123 

4-723 

30 

In  a  similar  way,  positive  values  have  been  assigned  to  the 
exercises  of  each  of  the  other  tests.  The  data  and  results  are 
given  in  Tables  VII-XIV.  Tables  VII,  IX,  XI  and  XIII  show 
that  a  sufficient  number  of  pupils  has  been  tested  to  give  fairly 
constant  results  for  each  test. 


TABLE  IX. — Average  positive  scores  for  each  exercise  of  Test  C. 


Schools 

Exercises 

Number  of 
Pupils 

I 

II 

III 

IV 

VII 

77 

69 

63 

54 

30 

VIII 

56 

61 

45 

58 

578 

IX 

55 

60 

45 

57 

651 

X 

55 

59 

44 

58 

682 

XI 

54 

59 

44 

58 

710 

XII 

53 

57 

42 

57 

795 

XIII 

53 

57 

41 

57 

844 

XXVIII 

53 

56 

40 

55 

882 

XXXII 

53 

55 

40 

55 

908 

XXXIV 

53 

55 

41 

54 

993 

XXXVII 

53 

55 

42 

55 

1019 

XXXVIII 

53 

55 

42 

54 

1047 

TABLE  X. — Positive  values  assigned  to  each  exercise  of  Test  C. 


Exercise 

Average  Score 

Difference 
Between  Score 
and  50% 

Distance  in 
P.E.  from 
Median 

Distance  Above 
Zero-point 

Value 
Assigned 

I 
II 
III 
IV 

52.6 
54-8 
41.8 
54-4 

-  2.6 
-4-8 

+  8.2 

-4-4* 

-  0.097 
-  0.179 
+  0.307 
—  0.164 

4.503 
4.421 
4.907 
4-436 

25 
24 
27 
24 

FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY 


49 


TABLE  XL — Average  positive  scores  for  each  exercise  of  Test  D. 


Schools 

Exercises 

Number  of 
Pupils 

I 

II 

III 

V 

88 

68 

65 

88 

XXII 

85 

70 

63 

132 

XXIII 

83 

68 

57 

207 

XXIV 

86 

70 

54 

275 

XXV 

83 

7i 

5i 

350 

XXVI 

84 

66 

53 

678 

XXVII 

86 

71 

54 

827 

XXIX 

84 

70 

52 

900 

XLVI 

83 

68 

52 

•     mi 

TABLE  XII. — Positive  values  assigned  to  each  exercise  of  Test  D. 


Exercise 

Average  Score 

Difference 
Between  Score 
and  50% 

Distance  in 
P.E.  from 
Median 

Distance  Above 
Zero-point 

Value 
Assigned 

I 
II 
III 

83.3 
68.1 

52.4 

-33-3 
-  I8.I 
-     2.4 

-  1-432 
—  0.698 
—  0.089 

3.168 
3-902 
4.5H 

2? 
34 
39 

TABLE  XIII. — Average  positive  scores  for  each  exercise  of  Test  E. 


Schools 

Exercises 

Number  of 
Pupils 

I 

II 

III 

IV 

I 

98 

98 

68 

32 

114 

II 

98 

98 

62 

39 

IQI 

III 

98 

96 

59 

30 

733 

IV 

98 

96 

57 

3i 

773 

V 

98 

96 

58 

36 

865 

VI 

97 

95 

56 

33 

962 

VII 

98 

95 

55 

34 

992 

LV 

98 

95 

55 

35 

1036 

TABLE  XIV. — Positive  values  assigned  to  each  exercise  of  Test  E. 


Exercise* 

Average  Score 

Difference 
Between  Score 
and  50  % 

Distance  in 
P.E.  from 
Median 

Distance  Above 
Zero-point 

Value 
Assigned 

I 

97-7 

-47-7 

-  2.958 

1.642 

12 

II 

94-9 

-44.9 

-  2.425 

,2.175 

16 

III 

54-6 

-    4-6 

—  0.172 

4.428 

33 

IV 

35-0 

+  15-0 

+  0.571 

5.I7I 

39 

VI.  CRITICAL  EXAMINATION  OF  THE  TESTS 
Test  D  has  been  criticized  on  the  ground  that  pupils  will  not 
understand  what  is  to  be  done  with  the  "Other  known  facts." 
The  returns  from  this  test  show  that  such  criticisms  are  not 


50  INVESTIGATION   OF   CERTAIN   ABILITIES 

well  founded.  In  only  a  few  cases  did  the  pupils'  papers  show 
that  they  misunderstood  the  test.  All  such  papers  were  rejected. 
Also  some  teachers  have  suggested  that  in  Test  E  the  lines  drawn 
by  the  pupils  are  the  results  of  guessing  rather  than  thinking. 
While  no  doubt  some  pupils  did  guess,  there  is  evidence  that  this 
is  not  generally  true.  In  some  cases  the  pupils  drew  new  figures 
on  their  papers  and  tried  out  several  lines  before  drawing  the 
lines  in  the  printed  figure.  Some  of  the  pupils  indicated  the 
relation  of  parts  of  the  figure  by  numbering  the  angles  or  marking 
the  sides  in  some  way.  In  many  cases  the  pupils  drew  one  or 
more  incorrect  lines  in  the  figure,  then  erased  them  and  drew  the 
correct  line.  A  careful  examination  of  the  papers  showed  that 
56  per  cent,  of  the  pupils  left  some  of  these  evidences  of  thought 
on  their  papers.  Moreover  many  of  those  who  drew  the  correct 
line  and  left  no  other  evidence  of  their  thought  on  their  paper 
undoubtedly  had  a  definite  method  of  proof  in  mind. 


Test  A 

V 

TT"   in*        TV         v 

0                      1 

2                       3 

4                        5 

G 

Test  R 

I      II    III               IV 

0 

2                        3 

4                         5 

6 

Test  C  HIV      III 


J                        1                        2 
Test  D 

3456 
I                II            III 

Test  E  I  II ITT  IV 

0  ~J 3 3  ~~I 5  6 

FIG.  2.     Linear  projection  of  the  difficulty  of  the  exercises  as  shown  in  Tables  VI, 

VIII,  X,  XII,  XIV. 

The  time  required  for  the  solution  of  a  single  exercise  has 
made  it  impossible  to  include  a  large  number  of  exercises  in 
each  test.  A  larger  number  of  carefully  selected  exercises  would, 
undoubtedly,  make  possible  a  more  accurate  discrimination 
between  varying  degrees  of  ability.  However,  this  difficulty  is 
not  as  great  as  it  at  first  appears;  for,  with  the  exception  of 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY 


Test  E,  each  exercise  consists  of  a  number  of  steps  each  of  which 
may  be  considered  as  a  separate  exercise,  just  as  the  words  in  a 
sentence  may  be  used  as  separate  elements  in  a  spelling  test. 
The  real  difficulty  here  lies  in  the  fact  that  we  have  not  been 
able  to  evaluate  these  steps  separately.1 


250- 

225- 
200- 

J1 

175" 

150- 
125- 

1 

c 

loo- 

75 

r-T 

50- 

-T 

— 

-, 

25- 

L 

0 

'  —  i 

0123456 


8    9  10  11  12 


FIG.  3.     Distribution  given  by  Exercise  I,  Test  C. 

Note. — The  vertical  line  x  marks  off  approximately  the  upper  ten  per  cent,  of  the 

class. 

Figure  2  is  the  linear  projection  of  the  difficulties  of  the  exer- 
cises as  shown  under  "Distances  above  zero-point"  in  Tables 
VI,  VIII,  X,  XII,  XIV.  The  exercises  of  the  different  tests  do 
not  begin  at,  or  extend  to,  the  same  points  on  the  scale.  Nor 
are  they  distributed  in  the  same  manner  over  the  portion  of  the 
scale  which  they  do  occupy.  Hence  the  tests  will  not  measure 


100- 
75- 
50- 
25- 

0 


FIG.  4.     Distribution  given  by  Exercise  II,  Test  C. 
Note.  —  The  vertical  line  x  marks  off  approximately  the  upper  ten  per  cent,  of  the 

class. 
1  Page  44. 


52  INVESTIGATION   OF   CERTAIN   ABILITIES 

the  respective  abilities  in  the  same  manner,  and  therefore  the 
results  obtained  from  the  different  tests  can  not  be  compared. 
Also  the  tests  would  be  more  satisfactory  if  the  exercises  were 


250- 

225- 

- 

200- 

£ 

J 

175- 

150- 

125- 

-n 

100- 

75- 

— 

Lr- 

50- 

25- 

L-, 

0 

l—J—  1  —  r 

01234    56    789  10  11  12  13 

FIG.  5.     Distribution  given  by  Exercise  III,  Test  C. 
Note. — The  vertical  line  x  marks  off  approximately  the  upper  ten  per  cent,  of  the 

class. 

distributed  over  a  larger  portion  of  the  scale  and  separated  by 
more  nearly  equal  intervals.  In  this  respect  Test  C  demands 
special  attention.  The  exercises  are  apparently  of  almost  the 


0    1    2    3    4    5   6    7    8   9  10  11 12  13  14  13  10  17  18  19  20  21  22  23  21 


FIG.  6.     Distribution  given  by  Exercise  IV,  Test  C. 

Note. — The  vertical  line  x  marks  off  approximately  the  upper  ten  per  cent,  of  the 

class. 

same  degree  of  difficulty.  This  is  due,  however,  to  the  arbitrary 
way  in  which  the  number  of  facts  considered  as  a  perfect  answer 
was  selected.1  If  the  separate  exercises  gave  exactly  the  same 

1  Page  36. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


53 


distribution  of  pupils  and  we  had  considered  as  a  perfect  answer 
to  each  exercise  the  smallest  number  of  facts  given  correctly  by 
any  one  of  exactly  the  highest  ten  per  cent,  of  pupils  taking  the 
test,  the  different  exercises  would  have  presented  the  same  degree 
of  difficulty.  However,  as  Figs.  3  to  6  show,  the  separate  exer- 


250- 

226 

200- 

175- 

150- 


100- 
75- 
50 
25- 


0-10  10-20  20-30  30-40  40-50  50-60  60-70  70-80  80-90  90-109 
FIG.  7.     Distribution  given  by  the  positive  scores  of  Test  A. 


250- 

225- 

200- 

175- 

150- 

125- 

100- 

75- 

50- 

25- 

0 


e-10  10-20  20-30  30-40  40-50  50-60  60-70  70-80  80-90  90-100 
FIG.  8.     Distribution  given  by  the  positive  scores  of  Test  B. 

cises  did  not  give  the  same  distribution,  and,  as  previously  noted,1 
we  were  not  able  to  select  exactly  the  highest  ten  per  cent,  of 
the  pupils.  Hence  there  is  some  variation  in  the  amount  of 
difficulty  presented  by  the  different  exercises  of  this  test,  but 
that  variation  is  slight. 
1  Page  36. 


54 


INVESTIGATION   OF   CERTAIN   ABILITIES 


In  any  test  an  important  consideration  is  the  form  of  distri- 
bution which  it  gives.  Figures  7  to  1 1  represent  the  data  given 
in  Tables1  XXVIII  to  XXXII  and  give  the  distribution  according 
to  the  positive  scores  for  Tests  A,  B,  C,  D  and  E2  respectively. 


200- 

175- 

150 

125 

100- 

75 

50 

25 

0 


0-10  10-20  20-30  3040  40-50  56^0  60-70  70-8f  80-90  90-100 
FIG.  9.     Distribution  given  by  the  positive  scores  of  Test  C. 


250 

225 

200^ 

175 

150 

125 

100- 

75- 

50- 

25 


0-10   10-20  20-30  30-40  40-50  50-60  60-70  70-80  80-90  90-100 
FIG.  10.     Distribution  given  by  the  positive  scores  of  Test  D. 

The  curves  for  Tests  A  and  C  are  nearly  normal,  but  those  for 
Tests  B,  D,  and  E  are  badly  skewed  toward  the  high  end  of  the 
scale.  Figures  12  to  16  are  the  frequency  curves  for  the  negative 
scores  and  represent  the  data  of  Tables3  XXXIII  to  XXXVII. 

1  Pages  75-88. 

2  As  Test  E  grouped  the  pupils  at  only  a  few  points  of  the  scale  the  class-intervas 
in  Fig.  ii  has  been  made  twice  as  large  as  in  Figs.  7-10. 

3  Pages  89-93. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


55 


With  the  exception  of  the  curve  for  Test  A,  they  also  are  badly 
skewed  toward  the  high  end  of  the  scale. 

This  skewness  may  be  due  in  part  to  the  elimination  of  the 
poorer  pupils  throughout  the  elementary  school  and  the  first 
year  of  the  high  school,  giving  a  specially  selected  group  with 


460- 


400- 
375- 


300- 
375- 
250* 
225- 
200- 
175 
150- 
125 
100 
75 
50 


0-20        20-40       40-60       60-80      80-100 
FIG.  ii.     Distribution  given  by  the  positive  scores  of  Test  E. 

which  we  have  worked.  The  importance  of  this  factor  is  in- 
creased in  those  schools  in  which  there  had  been  a  promotion 
between  the  time  the  study  of  geometry  was  begun  and  the  time 
the  tests  were  given,  resulting  in  an  elimination  of  pupils  who 
had  begun  the  study.1 

However,  a  more  important  cause  of  the  skewness  is  the 
selection  of  the  exercises.  Tests  B,  D  and  E  are  somewhat  too 
easy.  Also  there  are  too  few  exercises  in  Test  E  and,  as  we 
have  seen,  they  are  not  distributed  at  equal  intervals  along  the 
scale.  So  far  as  the  positive  scores  of  this  test  are  concerned  the 

»  Page  19. 


INVESTIGATION   OF  CERTAIN   ABILITIES 


exercises  do  not  admit  of  partial  answers.  Hence,  with  only  four 
exercises  not  more  than  fifteen  different  scores  are  possible.  In 
fact  Table  XXXIII  shows  that  the  exercises  are  such  that  the 
pupils  are,  for  the  most  part,  grouped  at  four  points  of  the  scale. 


150- 

125- 

100- 

75- 

50- 

25- 


17  16  15  14  13  i2  11 10  9    8    7  6    5   4    3   2    1    0 
FIG.  12.     Distribution  given  by  the  negative  scores  of  Test  A. 
250- 

225- 
200- 
175- 
150- 
125- 
100 

75 

50 

25 
0 


J5  U  1312  1110  9    8    7    6    5    4    3   2   1    0 
FIG.  13.     Distribution  given  by  the  negative  scores  of  Test  B. 

Evidently  two  exercises  selected  so  as  to  fall  at  equal  intervals 
between  exercises  II  and  III  would  give  more  satisfactory  results. 
Some  of  the  pupils  who  answered  exercises  I  and  II  but  could 
not  answer  III  or  IV  would  be  able  to  answer  one  or  both  of 
these  new  exercises.  Also  some  of  those  who  answered  either 
one  or  both  of  III  and  IV  would  fail  to  answer  one  of  the  new 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


57 


exercises.  Hence  more  pupils  would  be  grouped  along  the  central 
portion  of  the  scale  and  fewer  at  the  ends.  The  exercises  of  the 
other  tests  admit  of  partial  answers  and  therefore  this  same 


24  23  22  '2l'20'l9'l8'l7'l61151U>13i121iri01  9  V  7  '«  '  5  '1 '  3  '2  '  1 '  0 
FIG.  14.     Distribution  given  by  the  negative  scores  of  Test  C. 
300- 


275- 
250- 
225- 
200- 
175- 
150 
125 
100- 
75- 
50- 


o  o»  oo  t»  «  je  jjj  eg  «»  »-«  o  o»  «  t-.  «p  «5  ^  co  »«  i-* 
FIG.  15.     Distribution  given  by  the  negative  scores  of  Test  D. 

difficulty   does    not    arise    in    connection    with    them.     Tables 
XXVIII  to  XXXI  show  that  there  is  no  pronounced  tendency 
to  group  pupils  at  a  few  points  of  the  scale. 
5 


INVESTIGATION   OF  CERTAIN   ABILITIES 


When  considering  the  skewed  distributions  we  must  remember 
that  neither  the  positive  nor  negative  score  is  complete  in  itself. 
If  these  scores  could  be  combined  the  curve  would  be  moved 
toward  the  lower  end  of  the  scale  and  the  skewness  would  be 
decreased.  That  is,  if  a  pupil's  positive  and  negative  scores 
could  be  combined,  the  result  would  be  the  same  as  the  positive 


225 

200- 

175 

150- 

125- 

100- 

75 

50- 

25- 


m  nil  i  .  i  i  i  .  . 

13  1211  10  9    8    7654    32    10 


FIG.  1 6.     Distribution  given  by  the  negative  scores  of  Test  E. 

score  only  when  the  negative  score  is  zero.  In  all  other  cases  it 
would  be  less  and  he  would  take  a  lower  position  on  the  scale. 
It  may  be  argued  that  those  pupils  who  made  a  high  positive 
score  will  make  a  negative  score  near  zero  and  conversely,  leaving 
their  position  on  the  scale  practically  unchanged.  This,  however, 
has  been  shown  not  to  be  the  case.1 

Furthermore,  in  this  study  we  are  not  so  much  concerned 
with  the  exact  measure  of  pupils'  abilities  as  we  are  with  their 
ranks  when  arrayed  according  to  their  abilities.  If  our  tests 
enable  us  to  say  that  one  pupil  is  better  than  another  without 
saying  how  much  better,  our  purpose  will  be  served.  That  this 
condition  is  satisfied  by  Tests  A,  B,  C  and  D  is  indicated  in 
Tables  XXVIII  to  XXXI  by  the  fact  that  there  is  not  a  strong 
tendency  to  group  pupils  around  a  few  points  of  the  scale.  Hence 

1  Pages  24-27. 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY 


59 


we  may  conclude  that,  for  our  purpose,  Tests  A,  B,  C  and  D  are 
fairly  satisfactory,  but  that  Test  E  is  not.  The  data  of  Test  E 
will  be  included  in  this  study  to  throw  whatever  light  it  may 
on  our  conclusions  and  to  indicate  a  possible  field  for  further 
investigation.1 

VII.  EXAMINATION  OF  SCHOOL  GRADES 

Since  our  purpose  is  to  compare  the  results  of  the  tests  with 
the  pupils'  school  grades  we  shall  investigate  the  reliability  of 

1500- 


uoo- 

1300- 
1200- 
1100 
1000 
900- 
800- 
700- 
600- 
500- 
400- 
300 
200- 
100- 
0 


1  2  3  i  5 

FIG.  17.     Distribution  of  5195  pupils  given  by  their  school  grades. 

the  school  grades  as  shown  by  the  distribution  which  they  give. 
There  is  a  great  variation  in  the  part  of  the  scale  used  by  different 
schools.  One  school  may  give  grades  between  65  and  100  while 
another  gives  them  between  40  and  80.  Although  the  validity 

1  The  author  plans  to  improve  Test  E  and  make  further  investigation  with  it 
in  the  near  future. 


6o 


INVESTIGATION  OF  CERTAIN  ABILITIES 


TABLE  XV. — Number  of  pupils  falling  within  each  interval  into  which  the  part  of  the 
scale  used  by  each  school  is  divided. 


Schools 

I 

2 

3 

4 

5 

Totals 

I 

22 

46 

14 

27 

5 

114 

II 

7 

24 

16 

21 

9 

77 

III 

4 

12 

130 

277 

119 

542 

IV 

2 

4 

5 

17 

12 

40 

V 

7 

5 

29 

29 

22 

92 

VI 

14 

39 

o 

30 

U 

97 

VII 

3 

5 

12 

7 

3 

30 

VIII 

8 

94 

245 

103 

98 

548 

IX 

21 

22 

II 

16 

3 

73 

X 

5 

4 

9 

9 

4 

3i 

XI 

6 

3 

12 

6 

i 

28 

XII 

2 

i 

24 

35 

23 

85 

XIII 

16 

9 

II 

6 

7 

49 

XIV 

I 

5 

5 

5 

3 

19 

XV 

2 

3 

i 

8 

I 

15 

XVI 

27 

3 

12 

5 

4 

Si 

XVII 

I 

7 

9 

ii 

8 

36 

XVIII 

4 

3 

22 

55 

62 

146 

XIX 

29 

21 

7 

7 

13 

77 

XX 

I 

3 

93 

54 

26 

177 

XXI 

i 

17 

37 

0 

17 

72 

XXII 

6 

12 

17 

7 

2 

44 

XXIII 

5 

21 

49 

57 

31 

163 

XXIV 

3 

3 

10 

27 

25 

68 

XXV 

24 

37 

35 

62 

55 

213 

XXVI 

196 

68 

23 

34 

7 

328 

XXVII 

I 

33 

68 

30 

17 

149 

XXVIII 

4 

6 

8 

ii 

9 

38 

XXIX 

2 

12 

39 

15 

5 

73 

XXX 

2 

3 

8 

28 

13 

54 

XXXI 

6 

ii 

28 

7 

i 

53 

XXXII 

i 

5 

o 

16 

4 

26 

XXXIII 

4 

o 

8 

o 

3 

15 

XXXIV 

3 

12 

33 

21 

16 

85 

XXXV 

I 

2 

6 

IO 

4 

23 

XXXVI 

3 

2 

3 

II 

7 

26 

XXXVII 

i 

3 

4 

9 

9 

26 

XXXVIII 

ii 

4 

6 

2 

5 

28 

XXXIX 

6 

7 

0 

3 

i 

17 

XL 

6 

18 

12 

7 

4 

47 

XLI 

2 

6 

13 

II 

16 

48 

XLII 

6 

10 

14 

13 

8 

Si 

XLI  1  1 

I 

4 

8 

19 

21 

53 

XLIV 

5 

8 

IS 

7 

17 

52 

XLV 

6 

7 

8 

13 

17 

5i 

FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


61 


TABLE  XV. — Continued. 


Schools 

z 

2 

3 

4 

5 

Totals 

XLVI 

14 

56 

43 

59 

39 

211 

XLVII 

i 

7 

9 

13 

6 

36 

XLVI  1  1 

2 

9 

13 

12 

II 

47 

XLIX 

4 

IS 

21 

27 

19 

86 

L 

I 

2 

3 

5 

3 

14 

LI 

3 

2 

3 

4 

2 

14 

LII 

2 

6 

II 

22 

7 

48 

LIII 

3 

6 

18 

27 

17 

71 

LIV 

I 

o 

9 

6 

6 

22 

LV 

I 

3 

12 

9 

19 

44 

LVI 

6 

6 

9 

23 

19 

63 

LVII 

2 

6 

9 

13 

8 

38 

LVIII 

7 

ii 

17 

9 

8 

52 

LIX 

I 

4 

5 

14 

5 

29 

LX 

3 

5 

5 

10 

6 

29 

LXI 

10 

3 

2 

3 

7 

25 

LXII 

43 

39 

34 

27 

7 

ISO 

LXIII 

6 

IS 

23 

30 

14 

88 

Totals  .  . 

506 

810 

136s 

1461 

<KA 

SICK 

of  a  grading  system  does  not  depend  as  much  upon  the  part  of 
the  scale  used  as  it  does  on  the  accuracy  of  the  distribution 
along  that  part;  nevertheless,  in  order  to  compare  different 


100 1 
75 
50 
25- 

0 


1        2        345 
FIG.  18.     Distribution  of  the  146  pupils  of  school  XVIII  given  by  school  grades. 

grading  systems  and  to  combine  the  grades  from  different  schools 
it  is  necessary  to  eliminate,  so  far  as  is  possible,  any  such  vari- 
ation. In  order  to  do  this  the  part  of  the  scale  used  by  each 
school  has  been  divided  into  five  equal  intervals  and  the  pupils 
have  been  grouped  according  to  the  intervals  within  which  they 
fall.  Beginning  with  the  lowest  the  intervals  are  numbered  from 
i  to  5. 


62 


INVESTIGATION   OF  CERTAIN   ABILITIES 


Table  XV  shows  how  the  5,195*  different  pupils  tested  are 
distributed  according  to  their  school  grades.  The  distribution 
of  the  total  number  of  pupils  is  represented  graphically  in  Fig.  17. 


100- 

75- 

50- 

25 

0 


12345 
FIG.  19.     Distribution  of  the  73  pupils  of  school  XXIX  given  by  school  grades. 

While  this  roughly  approximates  a  normal  frequency  surface, 
there  is  a  decided  skewness  towards  the  higher  end  of  the  scale. 
This  may  be  due  in  part  to  the  elimination  of  the  poorer  pupils 
throughout  the  elementary  school  and  the  first  year  of  the 
high  school,  but  it  no  doubt  also  indicates  a  tendency  on  the 


100- 

75- 
50- 


1        2        345 
FIG.  20.     Distribution  of  the  52  pupils  of  school  LVIII  given  by  school  grades. 

part  of  teachers  to  avoid  not  only  the  lower  end  of  the  scale, 
but  also  the  lower  end  of  that  portion  of  the  scale  which  they 
use.  A  study  of  Table  XV  reveals  a  decided  variation  in  the 
form  of  distribution  given  by  the  grades  of  the  different  schools. 
Figures  18  to  23  represent  separately  the  distributions  of  the 
pupils  of  six  schools.  Figure  18  is  decidedly  skewed  towards  the 
higher  end  of  the  scale  while  Fig.  21  is  skewed  towards  the  lower 

1  The  sum  of  the  totals  given  in  Tables  V,  VII,  IX,  XI,  XIII  and  the  150  pupils 
from  school  LXII  not  included  in  these  totals  is  5,3 13-  This  apparent  inconsistency 
is  due  to  the  fact  that  88  pupils  of  school  V  and  30  pupils  of  school  VII  took  two 
tests,  making  118  duplications  in  the  total  5.313  pupils. 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY 


end.     Undoubtedly  these  distributions  are  not  based  accurately 
on  the  pupils'  abilities.     If  we  were  to  draw  the  frequency  surfaces 

100 
75- 
50 
25 

'     1        2    '     3   '     4        5 
FIG.  21.     Distribution  of  the  150  pupils  of  school  LXII  given  by  school  grades. 

100- 

75- 

50 

25- 

0 


123         4        5 
FIG.  22.     Distribution  of  the  97  pupils  of  school  VI  given  by  the  school  grades. 

V- 
200- 


175- 

150- 

125- 

100 

75 

50 

25- 

0 


12345" 
FIG.  23.  Distribution  of  the  328  pupils  of  school  XXVI  given  by  school  grades. 

for  each  of  the  sixty-three  schools  given  in  Table  XV,  we  would 
find  various  forms  of  distribution  lying  between  these  two 
extremes.  Some  of  these  are  approximately  normal  as  is  shown 


64  INVESTIGATION   OF   CERTAIN   ABILITIES 

by  Figs.  19  and  20.  Others  represent  peculiar  irregularities. 
Figure  22  gives  a  distribution  of  97  pupils  in  which  no  case  falls 
within  the  middle  interval,  a  condition  which  could  scarcely  exist 
if  this  number  of  pupils  were  distributed  accurately  according  to 
their  abilities.  Figure  23  gives  even  a  more  strikingly  irregular 
distribution.  In  this  case  60  per  cent,  of  the  pupils  fall  within 
the  lowest  interval.  In  fact  74  of  the  328  pupils,  or  more  than 
22  per  cent.,  received  the  lowest  passing  mark. 

Thus  it  is  evident  that,  even  where  a  fairly  large  group  of  pupils 
is  involved,  the  school  grades  frequently  do  not  give  a  normal 
distribution  and  they  very  probably  are  not  reliable  measures  of 
the  pupils'  abilities.  We  shall  now  investigate  the  relation  of  the 
test  and  school  grades. 

VIII.  COMPARISON  OF  SCHOOL  AND  TEST  GRADES 

As  noted  on  page  7  a  teacher's  grades  should  be  a  measure  of 
those  abilities  which  she  considers  of  value.  This,  of  course, 
will  not  always  be  the  case.  No  doubt  these  grades  are  often  a 
measure  of  the  pupil's  ability  to  memorize  and  reproduce  a  page 
of  geometry  text,  although  the  teacher  would  scarcely  admit 
that  such  is  the  case.  I,f  the  abilities  with  which  this  study  is 
concerned  are  among  those  which  the  teacher  considers  of  value, 
then  there  should  be  a  positive  correlation  between  the  test  and 
the  school  grades.  This  correlation  should  not  be  perfect  for 
there  are  several  abilities  involved  in  the  study  of  geometry  and 
there  should  not  be  a  perfect  correspondence  between  grades 
based  on  a  number  of  abilities  and  those  based  on  only  one  of 
these  abilities.  To  the  degree  that  the  tests  measure  the  four 
abilities  in  question,  the  coefficients  of  correlation  will  be  an 
index  of  one  of  two  things;  namely,  the  extent  to  which  the 
teacher  considers  these  abilities  of  value,  or  the  extent  to  which 
she  has  been  able  to  base  her  grades  on  the  abilities  which  she 
believes  to  be  of  value. 

Method  of  Determining  the  Correlation. — Since  it  has  been 
impossible  to  combine  the  positive  and  negative  scores  for  the 
different  tests,  it  is  impossible  to  compute  the  coefficient  of  corre- 
lation by  a  method  which  requires  actual  measures  of  abilities. 
If,  however,  the  individuals  can  be  arranged  in  a  series  according 
to  one  trait  and  then  rearranged  according  to  a  second  trait  the 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY        65 

correlation  between   the   two   traits  may  be  obtained   by  the 
formula 

p  '' 


N(N*  -  i) 

where  p  is  the  coefficient  of  correlation,  d  is  the  difference  between 
an  individual's  rank  in  one  series  and  his  rank  in  the  other  series, 
and  N  is  the  number  of  individuals  considered.1  However,  this 
formula  assumes  the  difference  between  any  two  successive  ranks 
to  be  the  same  for  all  parts  of  the  scale,  an  assumption  which  is 
not  true.  To  correct  this  error  Professor  K.  Pearson  has  de- 
veloped the  formula, 

r  =  2sin(|P) 
where 


p  =  i  - 


N(N2  - 


and  r  is  the  true  coefficient  of  correlation.  The  probable  error 
is  given  by  the  formula, 

0.7063(1  -  r2) 

JN 

This  means  that  it  is  an  even  chance  that  the  true  coefficient  of 
correlation  falls  within  the  limits  r  ±  P.E.  and  the  chances  are 
16  to  i  that  the  true  coefficient  will  fall  within  the  limits  r  ±  3P.E. 
That  is,  we  may  be  fairly  sure  that  there  is  a  positive  correlation 
when  r  >  3P.E.  This  method  of  computing  the  coefficient  of 
correlation  will  be  suited  to  our  purpose  if  we  can  rank  the 
pupils  according  to  the  school  grades  and  again  according  to  the 
test  scores.  But  we  encounter  a  difficulty  here  from  the  fact 
that  each  pupil  has  two  test  scores.  The  following  plan  has  been 
adopted  in  all  cases.  The  pupils  of  each  school  were  arranged 
according  to  their  positive  test  scores.  Beginning  with  the 
poorest  pupil,  they  were  numbered  from  i  upward.2  If  two  or 
more  pupils  tied  for  the  same  places,  the  sum  of  the  numbers 
belonging  to  these  places  was  distributed  equally  among  them. 
In  a  similar  way  numbers  were  assigned  according  to  the  negative 
scores,  the  higher  number  always  being  assigned  to  the  better 
pupil.  The  two  numbers  thus  assigned  to  each  pupil  were  then 

1  For  a  discussion  of  this  method  of  computing  the  coefficient  of  correlation  see 
William  Brown,  "The  Essentials  of  Mental  Measurement,"  pp.  42-53. 

2  It  is  to  be  noted  that  this  reverses  the  usual  order  in  which  ranks  are  assigned. 


66  INVESTIGATION   OF  CERTAIN   ABILITIES 

added,  and  the  pupils  were  again  ranked  according  to  these 
sums.  These  final  ranks  were  used  to  compute  the  coefficient 
of  correlation  between  the  school  and  the  test  marks.  This  gives 
equal  weight  to  the  positive  and  negative  scores,  and  we  are  not 
able  to  prove  that  this  is  as  it  should  be.  However,  it  is  probably 
as  accurate  as  any  other  method  of  combining  the  two  elements. 

Method  of  Dealing  with  the  Data  from  Different  Schools.— 
If  the  data  from  the  various  schools  could  be  combined  for  each 
test  we  would  gain  the  advantage  of  a  single  measure  of  corre- 
lation for  that  test.  However,  this  would  cause  us  to  lose  sight 
of  the  peculiarities  of  individual  schools.  Moreover  such  a 
procedure  is  impossible  because  of  the  great  variation  in  the 
grading  systems  of  the  various  schools.  A  pupil  marked  75  in 
one  school  may  be  a  better  student  than  one  marked  90  in  a 
second  school.  Unless  there  is  some  means  of  reducing  these 
grades  to  the  same  basis  it  would  be  impossible  to  arrange  the 
pupils  of  the  different  schools  in  a  single  series  according  to  their 
school  grades.  Hence  in  this  study  the  schools  have  been  dealt 
with  separately.  The  several  coefficients  of  correlation  will 
indicate  the  general  tendency  in  a  cumulative  way  and,  at  the 
same  time,  reveal  the  differences  in  the  practices  of  the  several 
schools.  It  may  be  argued  that  there  is  also  a  variation  in  the 
grading  systems  of  individual  teachers.  This  is  true,  but  usually 
the  variation  is  not  so  great  in  the  case  of  teachers  in  the  same 
system  as  it  is  in  the  case  of  different  schools.  Constant  inter- 
course among  the  teachers  and  other  influences  within  the  school 
tend  to  unify  the  standards  of  a  department.  However,  the 
fact  remains  that  there  is  a  variation  in  the  teachers'  standards 
and  this  to  a  certain  extent  weakens  our  conclusions. 

The  Coefficients  of  Correlation. — Space  does  not  permit  of  a 
complete  statement  of  the  computation  of  the  coefficient  of 
correlation  for  each  of  the  sixty-three  schools.  The  work  for 
one  school  is  given  in  detail  below.  Only  the  results  are  given 
for  the  other  schools. 

The  pupils  of  school  XXXVI  took  Test  A.  The  differences 
between  their  ranks  according  to  their  test  and  school  grades 
and  the  method  of  obtaining  these  differences  are  given  in 
Table  XVI.1  In  Table  XVII,  d  is  the  difference  between  test 

1  The  numbers  of  the  first  column  replace  the  pupils'  names  and  have  no  rela* 
tion  to  their  ranks. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


TABLE  XVI. — Differences  between  ranks  according  to  test  and  school  scores. 


Pupil 

Rank 
According 
to  +  Scores 

Rank 
According 
to  —  Scores 

Sum  of 
+  and 
—  Ranks 

Rank 
According  to 
Sum  of  +  and 
-  Ranks 

Rank 
According  to 
School  Grades 

Difference 
Between 
Test  and 
School  Rank 

I 

14-5 

13-5 

28.0 

13-0 

24-5 

II.  5 

2 

5-0 

8.0 

13-0 

4.0 

17-5 

13-5 

3 

6-5 

13-5 

2O.  O 

7-5 

17-5 

10.0 

4 

I.O 

5-o 

6.0 

2.0 

8.0 

6.0 

5 

16.0 

2.0 

18.0 

5-o 

22.O 

17.0 

6 

24.0 

22.0 

46.0 

25.0 

22.  0 

3-0 

7 

23.0 

9-5 

32-5 

19.0 

17-5 

1-5 

8 

26.0 

22.0 

48.0 

26.0 

24-5 

i-5 

9 

8-5 

22.0 

30-5 

17.5 

I4.O 

3-5 

10 

25.0 

18.5 

43-5 

24.0 

26.O 

2.O 

ii 

20.  0 

13-5 

33-5 

20.O 

22.0 

2.0 

12 

2O.O 

9-5 

29-5 

16.0 

I4.O 

2.0 

13 

2O.O 

22.O 

42.0 

23.0 

6.0 

'     17-0 

14 

10.5 

18.5 

29.0 

14.5 

4-5 

10.0 

IS 

22.O 

3-0 

25.0 

I  I.O 

14.0 

3-0 

16 

18.0 

17-0 

35-0 

2I.O 

2-5 

18.5 

17 

2.0 

I.O 

3-0 

I.O 

2-5 

i.S 

18 

10.5 

13.5 

24.0 

9-5 

7.0 

2.5 

19 

17.0 

7.0 

24.0 

9-5 

9-5 

o.o 

20 

3-0 

26.0 

29.0 

14-5 

I.O 

13.5 

21 

13.0 

25-0 

38.0 

22.0 

9.5 

12.5 

22 

8-5 

22.0 

30.5 

17.5 

ii.  5 

6.0 

23 

4.0 

5-0 

9.0 

3-0 

4-5 

i-5 

24 

14-5 

5-0 

19-5 

6.0 

n-5 

5-5 

25 

12.0 

13.5 

25-5 

12.0 

20.  o 

8.0 

26 

6-5 

13.5 

2O.O 

7-5 

17-5 

'        10.  0 

and  school  ranks,  K  is  the  number  of  times  each  difference  occurs 
in  Table  XVI,  and  2d2  is  the  sum  of  the  squares  of  the  differences. 

TABLE  XVII. — Sum  of  the  squares  of  the  differences  between  test  and  school  ranks. 


d 

K 

Kd* 

d 

K 

Kd* 

O.O 

I 

0.00 

8.0 

I 

64.00 

i-5 

4 

9.00 

10.0 

3 

300.00 

2.O 

3 

12.00 

ii-S 

I 

132.25 

2-5 

I 

6.25 

12.5 

I 

156.25 

3-o 

2 

18.00 

13-5 

2 

364.50 

3-5 

T 

12.25 

17.0 

2 

578.00 

5-5 

I 

30.25 

18.5 

I 

342.25 

6.0 

2 

72.0O 

Srf2  =  2097.00 

Substituting  N  =  26  and  2d2  =  2097  in  the  formula 


68 


INVESTIGATION   OF   CERTAIN   ABILITIES 


I    — 


we  get 

p  =  0.28. 

Correcting  this  result  by  the  formula 


we  get 
The  formula 


P.E.  = 


r  —  0.292. 

0.7063(1  -  r2) 


gives 


P.E.  =  0.127. 


The  coefficient  is  less  than  three  times  the  probable  error.  Hence 
in  the  case  of  school  XXXVI  the  pupils'  ability  to  draw  a  figure 
for  a  theorem  as  measured  by  Test  A  has  but  slight,  if  any, 
relation  to  the  school  grades  which  they  received. 

TABLE  XVIII. — Coefficients  of  correlation  for  Test  A. 


School 

r 

P.E. 

Relation  of  r  to  3  P.E. 

XXIII 

0.313 

0.068 

r  >  3  P.E. 

XXV 

0-395 

0.071 

r  >  3  P.E. 

XXXIII 

O.III 

0.180 

r  <  3  P.E. 

XXXV 

0.628 

0.089 

r  >  3  P.E. 

XXXVI 

0.303 

0.125 

r  <  3  P.E. 

XXXIX 

0.426 

0.140 

r  >  3  P.E. 

XL 

0.487 

0.079 

r  >  3  P.E. 

XLI 

0.303 

0.093 

r  >  3  P.E. 

XLII 

0.697 

0.051 

r  >  3  P.E. 

XLIII 

0.436 

0.088 

r  >  3  P.E. 

XLIV 

0.364 

0.085 

r  >  3  P.E. 

XLV 

0.364 

0.086 

r  >  3  P.E. 

L 

0.588 

0.123 

r  >  3  P.E. 

LI 

O.24O 

0.188 

r  <  3  P.E. 

LII 

0.528 

0.074 

r  >  3  P.E. 

LIII 

0.588 

0.055 

r  >  3  P.E. 

LIV 

—  0.150 

0.147 

r  <  3  P.E. 

LVI 

0.199 

0.085 

r  <  3  P.E. 

LIX 

0.688 

0.069 

r  >  3  P.E. 

LX 

0.578 

0.087 

r  >  3  P.E. 

LX1 

0.548 

0.099 

r  >  3  P.E. 

LXII 

0.436 

0.053 

r  >  3  P.E. 

LXIII 

0.292 

0.069 

r  >  3  P.E. 

FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


69 


A  study  of  Table  XVIII  shows  that  for  Test  A  the  coefficient  of 
correlation  varies  from  —  0.150  to  0.697.  F°r  x^  of  the  23  schools 
the  coefficient  is  greater  than  3P.E.  and  therefore  has  scientific 
significance.  For  schools  XXXV,  XLII,  L,  LII,  LIII,  LIX,  LX 
and  LXI  the  coefficients  are  probably  as  large  as  we  can  expect, 
if  we  remember  that  the  ability  to  draw  a  figure  is  only  one  of 
several  factors  upon  which  school  grades  in  geometry  depend. 
In  schools  XXXIII,  XXXVI,  LI,  LIV  and  LVI  there  seems  to 
be  but  little  relation  between  school  grades  and  the  ability  to 
draw  figures.  In  the  remaining  schools  the  positive  correlations 
are  but  slight.  In  school  LXI  I  the  test  was  given  after  all  of 
plane  geometry  had  been  completed  and  the  results  were  com- 
pared with  school  grades  given  for  the  first  two  books.  We 
would  expect  such  a  condition  to  reduce  the  coefficient  of  corre- 
lation. Nevertheless  Table  XVIII  shows  that  there  was  con- 
siderable relation  between  the  test  and  school  grades.  If  data 
were  at  hand,  it  would  be  interesting  to  determine  the  effect  of 
the  second  half-year  of  training  on  the  rank  of  pupils  as  deter- 
mined by  the  first  half-year  of  training. 

TABLE  XIX.— Coefficients  of  correlation  for  Test  B. 


School 

r 

P.E. 

Relation  of  r  to  3  P.E. 

XIV 

0.467 

0.127 

r  >  3  P.E. 

XV 

0.548 

0.127 

r  >  3  P.E. 

XVI 

0.447 

0.079 

r  >  3  P.E. 

XVII 

0.140 

0.115 

r  <  3  P.E. 

XVIII 

0.188 

0.056 

r  >  3  P.E. 

XIX 

0.230 

0.076 

r  >  3  P.E. 

XX 

0.219 

0.051 

r  >  3  P.E. 

XXI 

0.323 

0.075 

r  >  3  P.E. 

XXV 

0.568 

0.059 

r  >  3  P.E. 

XXX 

0-395 

0.081 

r  >  3  P.E. 

XXXI 

0.271 

0.090 

r  >  3  P.E. 

XLVII 

0.668 

0.065 

r  >  3  P.E. 

XLVIII 

0.261 

0.096 

r  <  3  P.E. 

XLIX 

0.538 

0.054 

r  >  3  P.E. 

LVI  I 

0.588 

0.075 

r  >  3  P.E. 

LVI  II 

0-344 

0.086 

r  >  3  P.E. 

Table  XIX  shows  that  the  coefficients  of  correlation  between 
the  scores  for  Test  B  and  the  school  grades  are  generally  low. 
For  14  of  the  16  schools  tested  the  coefficients  are  greater  than 


INVESTIGATION  OF  CERTAIN  ABILITIES 


3  P.E.,  and  therefore  it  is  quite  probable  that  there  is  a  positive 
correlation  in  these  cases.  For  schools  XV,  XXV,  XLVII, 
XLIX  and  LVII  the  coefficients  are  probably  as  large  as  can  be 
expected,  but  for  the  other  schools  the  correlation  is  low  and  in 
schools  XVIII  and  XLVII  I  there  is,  perhaps,  little  or  no  relation 
between  the  pupil's  ability  to  state  the  hypothesis  and  conclusion 
and  the  ability  upon  which  his  school  grade  is  based. 

TABLE  XX. — Coefficients  of  correlation  for  Test  C. 


School 

r 

P.E. 

Relation  of  r  to  3  P.E. 

VII 

0.209 

0.123 

r  <  3  P.E. 

VIII 

0.548 

0.030 

r  >  3  P.E. 

IX 

0.178 

0.080 

r  <  3  P.E. 

X 

0.385 

0.108 

r  >  3  P.E. 

XI 

0.416 

O.IIO 

r  >  3  P.E. 

XII 

0.538 

0.054 

r  >  3  P.E. 

XIII 

0.333 

0.090 

r  >  3  P.E. 

XXVIII 

0.436 

0.093 

r  >  3  P.E. 

XXXII 

0.042 

0.138 

r  <  3  P.E. 

XXXIV 

0.487 

0.058 

r  >  3  P.E. 

XXXVII 

0.406 

0.116 

r  >  3  P.E. 

XXXVIII 

0.508 

0.099 

r  >  3  P.E. 

Table  XX  shows  that  the  correlation  between  the  scores  for 
Test  C  and  the  school  grades  is  generally  low.  For  9  of  the  12 
schools  tested  the  coefficients  of  correlation  are  greater  than  3 
P.E.  and  therefore  there  is  very  probably  a  positive  correlation 
between  the  pupil's  ability  to  recall  facts  about  a  figure  and  his 
school  grade.  For  schools  VIII,  XII  and  XXXVIII  the  coef- 
ficients are  probably  as  large  as  can  be  expected,  but  for  the 
other  schools  they  are  low  and  in  schools  VII,  IX  and  XXXII 
there  is,  perhaps,  little  or  no  relation  between  a  pupil's  ability  to 
recall  geometrical  facts  and  his  school  grades. 

Table  XXI  shows  that  a  similar  condition  exists  for  Test  D. 
There  is  generally  a  low  positive  correlation  between  the  pupil's 
ability  to  select  and  arrange  facts  to  produce  a  proof  and  his 
school  grade.  Of  the  9  schools  tested  7  have  a  coefficient  of 
correlation  greater  than  3  P.E.  For  schools  XXII  and  XXIX 
the  coefficients  are  probably  as  large  as  can  be  expected,  but  for 
the  other  schools  they  are  small  and  in  schools  XXIII  and 
XXVII  there  is,  perhaps,  little  or  no  relation  between  the  test 
and  school  grades. 


FUNDAMENTAL    TO   THE    STUDY   OF   GEOMETRY 


Although  the  selection  of  exercises  for  Test  E  is  far  from  satis- 
factory, Table  XXII  shows  almost  as  favorable  results  as  were 
obtained  from  the  other  tests.  There  is  generally  a  low  positive 
correlation  between  the  test  and  school  grades.  Six  of  the  eight 
schools  tested  have  a  coefficient  greater  than  3  P.E.  and  there- 
fore there  is  very  probably  a  positive  correlation  between  the 
ability  to  draw  auxiliary  lines  and  the  abilities  upon  which 
school  grades  are  based.  The  coefficient  for  school  II  is  fairly 
large,  but  for  the  other  schools  it  is  usually  low,  and  in  schools 

TABLE  XXL— Coefficients  of  correlation  for  Test  D. 


School 

r 

P.E. 

Relation  of  r  to  3  P.E. 

V 
XXII 
XXIII 
XXIV 
XXV 

0.3SI 
0.528 
0.20Q 
0.325 
0-395 

0.066 
0.077 
0.078 
0.077 
O.OI3 

r  >  3  P.E. 
r  >  3  P.E. 
r  <  3  P.E. 
r  >  3  P.E. 
r  >  3  P.E. 

XXVI 
XXVII 
XXIX 
XLVI 

0.303 
0.126 
0.568 
0.323 

0.035 
0.057 
0.056 

o  044 

r  >  3  P.E. 
r  <  3  P.E. 
r  >  3  P.E. 
r  >  3  P.E. 

TABLE  XXII.  —  Coefficients  of  correlation  for  Test  E. 

School 

r 

P.E. 

Relation  of  r  to  3  P.E. 

I 
II 
III 
IV 
V 

0.216 

0.608 
0.139 
0.031 
0-493 

0.060 
0.051 
O.O29 
O.II2 
0.056 

r  >  3  P.E. 
r  >  3  P.E. 
r  >  3  P.E. 
r  <  3  P.E. 
r  >  3  P.E. 

VI 
VII 
LV 

0.253 
0.229 
0-343 

0.067 
O.I22 
O.O94 

r  >  3  P.E. 
r  <  3  P.E. 
r  >  3  P.E. 

IV  and  VII  there  is,  perhaps,  little  or  no  relation  between  the 
test  and  school  grades.  The  comparatively  favorable  results 
obtained  from  the  poorly  selected  exercises  of  Test  E  may  be 
due  to  the  fact  that  these  exercises  test  several  abilities  rather 
than  a  single  ability.  That  is,  if  it  is  true  that  a  pupil  must 
have  a  definite  proof  of  a  theorem  in  mind  before  he  can  draw 
the  proper  auxiliary  lines,  then  Test  E  will  measure  the  same 
abilities  that  Tests  C  and  D  measure  and  therefore,  other  things 
being  equal,  Test  E  should  give  the  highest  correlation. 


72  INVESTIGATION   OF   CERTAIN   ABILITIES 

Conclusion. — Among  the  different  schools  there  is  a  great 
variation  in  the  relation  between  the  test  and  school  grades. 
There  is  usually  a  positive  correlation  but  in  only  a  few  schools 
is  this  correlation  high.  In  some  of  the  schools  the  coefficient  is, 
perhaps,  affected  by  the  elimination  of  the  poorer  pupils.1  As 
it  is  easier  to  distinguish  the  extreme  cases,  the  elimination  of 
the  poorer  pupils  would  tend  to  reduce  the  correlation.  But 
if  a  correction  could  be  made  for  this,  it  is  quite  probable  that 
the  correlation  would  remain  low. 

Most  schools,  in  some  way,  emphasize  each  of  the  four  abilities 
which  this  study  investigates.  If  these  abilities  are  of  value  in 
themselves  or  if  they  furnish  a  basis  for  other  results  which  are 
of  value,  the  school  grades  should  bear  a  closer  relation  to  them. 
If,  on  the  other  hand,  the  coefficients  of  correlation  can  be  taken 
as  indices  of  the  values  of  these  abilities,  then  these  values  are, 
in  many  cases,  so  slight  that  the  schools  are  scarcely  justified  in 
giving  as  much  time  to  this  phase  of  geometry  as  is  now  given 
to  it. 

IX.  THE  EXTENT  TO  WHICH  THE  ABILITIES  ARE 
DEVELOPED 

Constancy  of  Results. — It  will  be  of  interest  to  see  the  extent 
to  which  the  schools  succeed  in  developing  each  of  the  four 
abilities.  For  this  purpose  the  median  scores  for  the  pupils  of 
each  school  and  for  all  the  pupils  have  been  computed.  How- 
ever, before  drawing  any  conclusions  from  the  combined  data  of 
the  different  schools  we  should  determine  whether  a  sufficient 
number  of  pupils  has  been  tested  to  eliminate  chance  variation. 
As  previously  noted  Tables  V,  VII,  IX,  XI  and  XIII  show  that, 
so  far  as  positive  scores  are  concerned,  this  condition  has  been 
fairly  well  realized.  Tables  XXIII  to  XXVII  give  the  average 
negative  scores  for  each  test  in  a  cumulative  way.  A  study  of 
these  tables  shows  that  a  fairly  constant  condition  has  been 
obtained  in  the  case  of  each  test.  The  results  of  Test  D  are 
less  satisfactory  in  this  respect  than  those  of  any  of  the  other 
tests. 

1  Page  19. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  73 

TABLE  XXIII. — Average  negative  scores  for  each  exercise  of  Test  A. 


Schools 

Exercises 

Number  of 
Pupils 

I 

II 

III 

IV 

V 

XXIII 

0.56 

0-54 

0-49 

3-84 

3-07 

88 

XXV 

0.66 

0.52 

0-57 

3.10 

i.  IS 

158 

XXXIII 

0.64 

0-54 

0.60 

3-01 

1.88 

173 

XXXV 

0.63 

0-54 

0.64 

2.91 

2.06 

196 

XXXVI 

0.60 

0-54 

0.70 

2-73 

2.12 

222 

XXXIX 

0.62 

0-53 

0.64 

2.62 

2.IO 

239 

XL 

0-59 

0.51 

0.71 

2.76 

2.IO 

286 

XLI 

0-53 

0.49 

0.68 

2.74 

2.O2 

334 

XLII 

0.49 

0-45 

0.68 

2.60 

2.12 

385 

XLIII 

0.46 

0.42 

0.65 

3.00 

2.  II 

438 

XLIV 

0.44 

0.41 

0.66 

2.88 

2.13 

490 

XLV 

0.41 

0.39 

0.63 

3-04 

2.25 

541 

L 

0.41 

0.39 

0.63 

3-03 

2-30 

555 

LI 

0.41 

0.38 

0.64 

3.02 

2.30 

569 

LII 

0.41 

0.40 

0.66 

2.87 

2.27 

617 

LIII 

0-43 

0.41 

0.70 

3-12 

2.38 

688 

LIV 

0.45 

0.41 

0.70 

3.09 

2-35 

710 

LVI 

0-45 

0.44 

0.71 

3.02 

2-34 

773 

LIX 

0-45 

0.44 

0.69 

3-01 

2.36 

802 

LX 

0-45 

0.44 

0.68 

3-12 

2.39 

831 

LXI 

0.44 

0.44 

0.68 

3-07 

2.30 

856 

LXIII 

0.44 

0.43 

0.71 

2.96 

2-39 

944 

TABLE  XXIV. — Average  negative  scores  for  each  exercise  of  Test  B. 


Schools 

Exercises 

Number  of 
Pupils 

I 

II 

III 

IV 

XIV 

0.16 

0.32 

1.83 

2.63 

19 

XV 

0.24 

0.29 

1.32 

1-97 

34 

XVI 

0.16 

0.21 

1.  12 

1-93 

85 

XVII 

0.14 

0.18 

0.94 

1.92 

121 

XVIII 

0.08 

0.18 

0.75 

2.24 

267 

XIX 

0.07 

0.17 

0.68 

2.13 

344 

XX 

0.09 

0.25 

0.71 

2.16 

521 

XXI 

O.IO 

0.24 

0.74 

2.13 

593 

XXV 

O.I2 

0.25 

0.80 

2.17 

659 

XXX 

0.13 

0.28 

0.84 

2.15 

713 

XXXI 

0.14 

0.28 

0.81 

2.17 

766 

XLVII 

0.14 

0.28 

0.90 

2.14 

802 

XLVIII 

0.13 

0.27 

0.89 

2.16 

849 

XLIX 

0.13 

0.27 

0.90 

2.15 

935 

LVII 

0.13 

0.27 

0.90 

2.16 

975 

LVI  1  1 

0.14 

0.28 

0.92 

2.16 

1025 

74 


INVESTIGATION   OF  CERTAIN   ABILITIES 


TABLE  XXV. — Average  negative  scores  for  each  exercise  of  Test  C. 


Schools 

Exercises 

Number  of 
Pupils 

. 

II 

III 

IV 

VII 

0.23 

0.23 

1.77 

1-93 

30 

VIII 

0.28 

0.58 

1.84 

2.O2 

578 

IX 

0.26 

0-59 

1.87 

2.08 

651 

X 

0.26 

0.58 

1.87 

2.OQ 

682 

XI 

0.26 

0.58 

1.84 

2.IO 

710 

XII 

0.28 

0.63 

1.81 

2.09 

795 

XIII 

0.29 

0.62 

1.78 

2.09 

844 

XXVIII 

0.30 

0.61 

1.76 

2.07 

882 

XXXII 

0.30 

0.60 

1.76 

2.04 

908 

XXXIV 

0.31 

0.65 

1.87 

2.14 

993 

XXXVII 

0.30 

0.66 

1.87 

2.14 

1019 

XXXVIII 

0.31 

0.65 

1.87 

2.07 

1047 

TABLE  XXVI. — Average  negative  scores  for  each  exercise  of  Test  D. 


Exercises 

Number  of 

Schools 

I 

II 

III 

Pupils 

V 

0.68 

.16 

1.02 

88 

XII 

0.77 

•  13 

1.22 

132 

XXIII 

1.05 

.28 

1-54 

207 

XXIV 

0.84 

.09 

1.42 

275 

XXV 

I.OI 

.14 

1.50 

350 

XXVI 

0.80 

.08 

1.49 

678 

XXVII 

o.93 

.00 

1.50 

827 

XXIX 

0.91 

.00 

i-4S 

900 

XLVI 

0.81 

0.91 

1.31 

IIII 

TABLE  XXVII. — Average  negative  scores  for  each  exercise  of  Test  E. 


Schools 

Exercises 

Number  of 
Pupils 

I 

II 

III 

IV 

I 

0-34 

0-33 

0.72 

1.22 

114 

II 

0.30 

0.28 

0.63 

1.05 

191 

III 

0.32 

0.24 

0.68 

1.07 

733 

IV 

0.33 

0.25 

0.68 

1.  06 

773 

V 

0-33 

0.27 

0.78 

0.98 

865 

VI 

0-35 

0.28 

0.70 

0-99 

962 

VII 

0.35 

0.27 

0.71 

0.96 

992 

LV 

0.36 

0.28 

0.71 

0.97 

1036 

Standards  of  Achievements.— Tables  XXVIII  to  XXXII  give 
the  number  of  pupils  receiving  each  positive  score  for  each  test. 
The  median  scores  for  each  school  and  for  all  pupils  tested  are 
also  given.  Similar  data  for  the  negative  scores  are  given  in 
Tables  XXXIII  to  XXXVII. 


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FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  83 

TABLE  XXX. — Number  of  pupils  receiving  each  positive  score  for  Test  C. 


_ 

M 

> 

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INVESTIGATION   OF  CERTAIN   ABILITIES 
TABLE  XXX.— Continued. 


Score 

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N 

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XXVIII 

XXXII 

AIXXX 

XXXVII 

XXXVIII 

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FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  85 

TABLE  XXX. — Continued. 


Score 

i—  i 
> 

> 

K 

X 

X 

S 

M 

XXVIII 

XXXII 

XXXIV 

> 

XXXVIII 

• 

1 

0 

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28 

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26 

28 

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54-1 

44.4 

46.3 

49.5 

38.8 

42-5 

29.0 

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55-2 

56.0 

49.0 

50.6 

2  ^  D6rcentile 

16.  <? 

75  percentile  . 

6-;  2 

Quartile 

14  4. 

86 


INVESTIGATION   OF  CERTAIN   ABILITIES 


TABLE  XXXI. — Number  of  pupils  receiving  each  positive  score  for  Test  D. 


Score 

V 

XXII 

XXIII 

XXIV 

XXV 

XXVI 

XXVII 

XXIX 

XLVI 

Totals 

O 

I 

I 

j 

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2 

2 

8 

2 

2 

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o 

12 

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14 

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17 

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18 

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5 

21 

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22 

I 

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2 

27 

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2 

24 

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26 

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27 

2 

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31 

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32 

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9 

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4 

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15 
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22 

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47 
48 

2 

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3 

2 

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3 

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5 
3 

7 

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39 
6 

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51 

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52 

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23 

54 

3 

2 

3 

4 

10 

5 

10 

37 

FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY 


TABLE  XXXI.— Continued. 


Score 

V 

XXII 

XXIII 

XXIV 

XXV 

XXVI 

XXVII 

XXIX 

XLVI 

Totals 

ee 

c 

5' 

56 

I 

4 

3 

8 

57 

2 

2 

i 

I 

2 

8 

eg 

2 

I 

7, 

CQ 

I 

3 

I 

6 

60 

i 

I 

II 

3 

I 

5 

22 

61 
62 

4 

i 

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3 

9 
I 

7 

i 

17 
6 

12 
2 

I 

2 

9 

62 
14 

67 

i 

I 

I 

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3 

7 

64 

i 

2 

2 

6 

3 

14 

6< 

2 

I 

i 

4 

66 
67 
68 

I 

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2 

4 
3 

I 

4 

4 

i 
6 

2 

4 
15 
4 

I 

3 
3 

2 

6 

12 
2 

18 

45 
19 

60 

I 

j 

i 

2 

2 

I 

8 

70 

3 

I 

2 

I 

2 

71 

2 

2 

T. 

2 

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7 

14. 

72 

I 

I 

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I 

71 

I 

2 

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19 

3 

6 

7.4. 

74 
75 

4 
4 

3 

2 

2 

5 

22 

I 

7 

6 

16 

67 
c 

76 

i 

7. 

7 

i 

8 

77 

i 

2 

I 

2 

i 

i 

I 

9 

78 

I 

i 

5 

i 

4 

12 

79 

i 

2 

i 

2 

6 

80 
81 

5 

i 

2 
-? 

6 

2 
2 

4 

2O 

3 

9 
5 

2 

4 
6 

52 

22 

82 

i 

i 

2 

83 
84 

4 
3 

I 

i 

5 

i 

i 

5 

4 

5 

2 

I 

3 

3 
3 

24 

18 

85 

i 

i 

I 

3 

86 

j 

i 

10 

I 

2  ^ 

87 
88 

II 

2 

i 

3 

3i 

12 

i 

10 

71 

o 

80 

i 

I 

7 

11 

QO 

o 

91 

4 

I 

3 

2 

6 

5 

i 

8 

30 

02 

7. 

i 

I 

2 

7 

93 
04 

i 

i 

3 

4 

12 

9 

i 

12 

4i 

2 

QC 

3 

3 

96 

2 

5 

2 

I 

IO 

97 

o 

08 

o 

99 

o 

100 

5 

4 

2 

5 

2 

25 

24 

3 

31 

101 

Totals  

88 

44 

75 

68 

75 

328 

149 

73 

211 

IIII 

Medians  .... 

75-0 

71-5 

66.4 

68.8 

61.6 

74-0 

80.5 

54-7 

74-7 

73-3 

25  percentile. 

55-4 

75  percentile. 

87.0 

Quartile  

15.8 

88 


INVESTIGATION   OF   CERTAIN   ABILITIES 


TABLE  XXXII. — Number  of  pupils  receiving  each  positive  score  for  Test  E. 


Score 

, 

II 

III 

IV 

V 

VI 

VII 

LV 

Totals 

o 

I 

I 

I 

o 

I  j 

O 

12 
I"? 

i 

i 

8 

I 

3 

5 

2 

21 

o 

iq 

O 

16 

i 

i 

6 

2 

IO 

17 

O 

27 

o 

28 

2O 

26 

20 

1  66 

13 

6 

56 

8 

8 

303 

o 

AA 

o 

AX 

12 

2 

2 

16 

46 

o 

48 

o 

40 

i 

c 

I 

7 

CQ 

o 

ci 

I 

I 

2 

4 

C2 

o 

C4 

o 

ere 

i 

I 

2 

q6 

o 

60 

o 

61 
62 

49 

16 

I98 

12 

5 

25 

5 

4 

314 

o 

66 

o 

67 
68 

9 

14 

52 

3 

25 

I 

12 

13 

129 
o 

71 

o 

72 

i 

I 

73 

o 

83 

o 

84 

j 

8« 

o 

87 

o 

88 

2 

2 

80 

o 

on 

o 

100 

27 

24 

88 

10 

48 

I 

5 

12 

215 

Totals  

114 

77 

542 

40 

92 

97 

30 

44 

1036 

Medians  

61.6 

62.0 

61.4 

61.4 

99.0 

28.7 

67.2 

67.3 

61.5 

25  percentile  .  . 

28.7 

75  percentile 

67  7 

Quartile  .  .  . 

IO  5 

FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


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FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  9! 

TABLE  XXXV. — Number  of  pupils  receiving  each  negative  score  for  Test  C. 


Score 

> 

> 

* 

X 

* 

£ 

>< 

XXVIII 

XXXII 

XXXIV 

XXXVII  1 

XXXVIII 

J2 
I 

O 
I 
2 
3 

4 

5 
6 

7 
8 

3 
5 
5 

2 

5 
3 

i 

85 

57 
68 
67 
62 

46 
29 
30 
15 

9 

7 

10 

6 

4 

2 

9 
4 
4 

7 
6 
6 
3 
3 

3 

i 
i 
i 

4 
3 
3 
4 
3 

3 

2 

I 
2 

12 
14 

8 
7 
9 

2 

4 
6 

T 

6 
6 
9 
6 

7 

i 

7 

i 

12 
3 

4 
5 
4 

I 

4 

I 
4 

2 

6 

S 

4 

2 

5 
5 
5 
5 

10 

3 
6 
13 

2 

3 

2 

5 
6 

2 
2 

3 

i 

5 
5 

i 

5 
4 

2 

4 

152 
117 
126 
122 
118 

69 
64 
67 
28 

9 

IO 

3 

13 

12 

3 

2 

2 

3 

4 

i 

i 

2 

I 

'  3 

4 

.... 

27 
27 

ii 

I 

12 

3 

i 

i 

I 

i 

I 

21 

12 

I 

14 

3 

4 

2 

A 

28 

13 
14 

5 

c 

3 

I 

4 

2 

I 

I 
I 

4 

i 

i 

20 

1C 

7 

4 

1  1 

16 

i 

c 

I 

i 

I 

17 

2 

2 

18 

4 

2 

6 

19 

2 

I 

20 

2 

2 

21 

I 

I 

2 

22 

o 

23 

3 

I 

24 

i 

I 

I 

•2 

2C 

•z 

26 

o 

27 

o 

28 

o 

29 

o 

•JO 

o 

H 

o 

32 

o 

33 

I 

I 

34 

I 

I 

-JIT 

i 

i 

2 

16 

i 

I 

Totals  

•2.O 

1:48 

7"? 

•21 

28 

8"? 

40 

18 

26 

8=? 

26 

28 

104.7 

Medians  

4.0 

4.0 

5-3 

2.4 

4.0 

4.2 

3-6 

3-o 

4-0 

7-3 

3-5 

3-6 

4.1 

25  percentile  .  . 

I  O 

75  percentile  .  . 

1 

7.1 

Quartile  

2.7 

92  INVESTIGATION   OF   CERTAIN   ABILITIES 

TABLE  XXXVI. — Number  of  pupils  receiving  each  negative  grade  for  Test  D. 


Score 

> 

H 

H 

X 

M 

> 

1 

> 
H 

X 

E 

XXVII 

H 

X 
X 

> 
»J 

X 

(A 

1 

O 

I 
2 
3 

4 

6 

7 

18 
17 
17 
9 
9 

6 
5 

2 

13 

3 

7 
4 
3 

5 

2 

ii 
15 
9 

7 
5 

6 

3 

4 

26 
14 
13 

4 

2 

2 
3 

3 

8 
13 
8 

13 
6 

6 

I 
4 

92 

57 
42 
29 
18 

24 
16 
ii 

44 
20 
13 
14 
8 

9 

5 

12 

25 
15 

3 
5 
ii 

3 

I 
3 

62 
28 

22 
23 

16 

17 

9 
ii 

299 
182 
134 
108 

78 

78 
45 
50 

8 
9 

IO 

2 

I 

I 
I 

2 

I 

I 

I 

2 

3 

2 

13 

4 

8 

4 
5 

e 

I 
3 

8 

5 

4 

33 
23 

21 

ji 

I 

I 

2 

2 

4 

i 

i 

12 

12 

2 

I 

I 

2 

2 

8 

13 
14 



2 

I 

I 
I 

4 

I 

I 
I 

2 

IO 

4 

je 

I 

I 

2 

I 

5 

16 

I 

2 

2 

I 

6 

17 

o 

18 

2 

I 

3 

10 

2 

2 

2O 

I 

I 

2 

21 

I 

22 

I 

27 

I 

24 

I 

2C 

I 

26 

I 

27 

o 

28 

o 

2O 

I 

I 

30 

o 

31 

o 

32 

o 

33 

o 

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o 

35 

o 

36 

o 

37 

I 

I 

Totals 

88 

44 

7  "? 

68 

7  "> 

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I4O 

7-1 

211 

I  II  I 

Medians  

1-5 

2.9 

2.4 

1.6 

3-7 

2.4 

2.8 

1.8 

2-7 

2.6 

25  percentile. 

O.Q 

75  percentile. 

tr.4 

Quartile  

23 

FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  93 

TABLE  XXXVII. — Number  of  pupils  receiving  each  negative  grade  for  Test  E. 


Score 

I 

II 

III 

IV 

V 

VI 

VII 

LV 

Totals 

O 
I 

2 

3 

4 

5' 

18 
14 
33 
17 
15 

8 

20 
19 
14 
13 

7 

2 

76 
117 

147 
92 
56 

24 

9 
6 

8 
4 
6 

? 

28 

21 
14 

16 

4 

4 

10 
19 

23 

18 

7 

10 

IO 

9 

4 

2 

4 

7 

IO 

3 
6 
6 

I78 
215 
246 
168 
105 

e6 

6 

7 

3 

5" 

I 

I 

16 
6 

I 

I 

4 

i 

3 
-i 

I 

3 

2 

32 

TO 

8 

2 

2 

T 

7" 

2 

I 

•I 

10 

I 

2 

I 

4 

j  j 

o 

12 

I 

i 

J-3 

I 

i 

2 

Totals  

114 

77 

542 

40 

92 

97 

30 

44 

1036 

Medians  

2.8 

2.0 

2-5 

2.6 

1.9 

2.8 

1.6 

3-3 

2-5 

25  percentile 

1-4 

75  percentile 

3.8 

Quartile  

1.2 

Conclusions. — Summaries  of  the  results  in  Tables  XXVII  to 
XXXVII  are  given  in  Tables  XXXVIII  and  XXXIX.  Since, 
as  previously  noted,  we  do  not  know  that  the  tests  measure  the 
respective  abilities  in  the  same  way,  we  can  not  compare  the 
results  of  the  different  tests.  Thus  the  low  median  positive 
score  for  Test  C  does  not  necessarily  indicate  that  the  pupils 
have  less  ability  to  recall  facts  about  a  figure  than  to  do  any 
one  of  the  things  called  for  in  the  other  tests.1  However  we  may 
compare  results  obtained  by  giving  the  same  test  in  different 
schools.  Such  a  comparison  shows  a  decided  variation  in  both 
the  positive  and  negative  scores  made  by  the  schools  taking  any 
one  of  the  tests.  In  the  case  of  each  test,  the  marks  of  some 
schools  are  quite  satisfactory  while  those  of  others  are  extremely 
low.  This  variation  in  achievement2  may  be  due,  in  part,  to 

1  The  low  median  scores  for  Test  C  are,  in  part,  due  to  the  arbitrary  selection 
of  the  number  of  facts  required  for  a  perfect  answer  to  each  exercise. 

2  The  high  maximum  score  for  Test  E  may  be  due  to  the  fact  that  the  last  exercise 
of  the  test  had  been  studied  in  class.     The  test  papers  indicate  that  this  might  be 
the  case  although  the  evidence  is  not  conclusive. 


94 


INVESTIGATION   OF   CERTAIN   ABILITIES 


TABLE  XXXVIII. — Summary  of  positive  scores. 


Test 

Medians  for  All 
Pupils  Tested 

25  Percentile 

75  Percentile 

Lowest  Median 
by  Any  School 

Highest  Median 
by  Any  School 

A 
B 
C 
D 
E 

62.5 
69.3 
50.6 
73-3 
61.5 

51-3 
51.8 
36.5 
55-4 
28.7 

72.9 
82.2 
65.2 
87.0 
67.5 

50.5 
38-5 
29.0 

54-7 
28.7 

78.7 
80.9 
67.0 
80.5 
99.0 

TABLE  XXXIX. — Summary  of  negative  scores. 


Test 

Median  for  All 
Pupils  Tested 

25  Percentile 

75  Percentile 

Lowest1  Median 
by  Any  School 

Highest  Median 
by  Any  School 

A 
B 
C 
D 
E 

7-1 
3-5 
4.1 
2.6 
2-5 

10.7 
5-9 
7-3 
5-4 
3-8 

4-2 
1-5 

1.9 

0.9 
1.4 

II.8 
4-5 
7-3 
3-7 
3-3 

4.8 

2.0 

2.4 

i-5 
1.6 

differences  in  local  conditions  rather  than  differences  in  methods 
and  in  teaching  ability.  Nevertheless,  it  is  difficult  to  see  how 
local  conditions  alone  could  result  in  the  extremely  low  scores 
made  by  some  of  the  schools.  If  the  abilities  tested  are  essential 
to  success  in  the  study  of  geometry,  then  the  results  indicate  that 
progress  is  almost  impossible  in  some  of  the  schools  until  these 
abilities  have  been  further  developed.  On  the  other  hand  the 
achievements  of  other  schools  indicate  that  it  is  altogether 
possible  to  develop  these  abilities  to  a  fair  degree  during  the 
study  of  the  first  two  books  of  geometry. 

School  LXII,  which  had  completed  plane  geometry  took  Test  A 
and  made  a  positive  score  of  71.5  and  a  negative  score  of  4.6. 
While  this  school  ranks  high  it  did  not  make  a  better  showing 
than  some  of  the  schools  which  had  completed  the  first  two 
books  only.  This  again  raises  the  question  of  the  effect  of 
further  training  such  as  is  now  given  in  our  schools. 

X.  USE  OF  THE  TESTS 

Thus,  although  it  is  possible  to  develop  the  abilities  with 
which  this  study  is  concerned,  some  schools  fail  to  do  so.  There- 
fore, if  these  abilities  are  essential  to  progress  in  geometry,  it  is 
important  that  we  have  some  means  of  determining  whether 
they  are  being  satisfactorily  developed  in  a  class.  Such  a 

1  As  the  negative  scores  represent  the  numbers  of  incorrect  and  unnecessary 
statements,  the  larger  numbers  represent  the  lower  scores. 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY       95 

diagnosis  will  enable  the  teacher  to  give  attention  to  the  par- 
ticular phases  of  the  subject  in  which  the  pupils  are  weak.  It  is 
believed  that  Tests  A,  B,  C  and  D  may  be  found  useful  for  this 
purpose.  As  nearly  as  possible,  each  of  these  tests  has  been 
arranged  to  measure  a  single  ability.  Moreover,  the  method  of 
grading  reveals  to  the  teacher  not  only  the  pupils'  positive  abili- 
ties but  also  the  extent  to  which  they  are  influenced  by  mis- 
conceptions. These  analytic  features  of  the  tests  are  of  im- 
portance, for  it  is  only  by  determining  the  elements  of  the  pupils* 
abilities  that  we  may  know  where  to  place  the  emphasis  in  our 
teaching. 

The  standards  given  in  Tables  XXXVIII  and  XXXIX  furnish 
a  means  of  comparison.  If  a  class  is  above  the  median  scores  in 
these  tables,  and  especially  if  it  is  near  or  above  the  75-percentile 
marks,  the  teacher  may  be  fairly  sure  that  the  abilities  have 
been  sufficiently  developed  to  insure  the  success  of  the  class. 
If,  however,  the  class  falls  below  the  median  score  in  any  one  of 
the  tests  and  especially  if  it  falls  near  or  below  the  25-percentile 
mark,  special  effort  should  be  made  to  develop  the  ability  in 
question. 

If  such  comparisons  are  to  be  trustworthy,  the  tests  should  be 
given  at  the  time  the  class  has  completed  the  first  two  books  of 
geometry,  and  the  rules  for  scoring  the  tests  given  on  pages 
97-99  of  the  appendix1  should  be  followed  carefully.  Further- 
more, it  should  be  remembered  that  these  tests  do  not  cover  the 
entire  field  of  geometry.  They  deal  with  abilities  essential  to 
the  formal  demonstration  of  theorems;  but  there  are  other 
phases  of  the  subject,  such  as  the  practical,  which  we  have  not 
investigated  and  it  is  possible  that  a  class  may  make  a  creditable 
showing  on  each  of  these  tests  and  yet  not  realize  the  greatest 
values  from  geometry.  Hence  teachers  should  not  rely  wholly 
upon  these  tests  as  a  means  of  determining  the  weaknesses  of 
their  classes. 

XL  CONCLUSIONS 

This  concludes  the  more  important  features  of  our  study.  Some 
minor  and  related  topics  will  be  discussed  in  the  appendix.  We 
have  assumed  that  the  abilities  investigated  are  essential  to  the 
study  of  geometry.  This  assumption  is  based  upon  long  experi- 
ence as  a  teacher  and  upon  practice  in  our  schools  all  of  which 

1  See  also  the  fuller  discussion  of  the  method  of  scoring  papers  on  pages  28-43- 


96  INVESTIGATION   OF   CERTAIN   ABILITIES 

emphasize  these  abilities  in  some  way  or  other.  In  some  cases 
the  school  grades  bear  a  close  relation  to  these  abilities,  but 
usually  this  relation  is  slight;  so  slight,  in  fact,  that  if  it  were 
a  true  index  of  the  value  of  these  abilities  the  time  spent  on  their 
development  could  not  be  justified.  However,  this  condition  is 
perhaps  due,  in  part,  to  the  teachers'  inability  to  grade  their 
pupils  accurately.  In  like  manner,  when  judged  by  the  scores 
made  on  any  one  of  the  tests,  the  schools  vary  greatly  in  their 
achievements.  While  the  returns  from  some  of  the  schools  are 
fairly  satisfactory,  in  many  cases  the  scores  are  so  low  as  to 
make  it  doubtful  whether  values  dependent  upon  these  abilities 
are  realized.  This  variation  in  achievement  may  be  due,  in  part, 
to  local  conditions;  but  it  is  doubtless  dependent,  to  a  certain 
extent,  upon  the  teachers'  efficiency,  which,  we  believe,  could  be 
increased  if  tests  similar  to  these  were  used  to  show  where 
emphasis  should  be  placed. 


APPENDIX 

For  the  benefit  of  any  who  care  to  give  the  tests  and  compare 
their  results  with  those  of  this  investigation  a  brief  statement  of 
directions  for  scoring  the  papers  is  given  below.  Also  certain 
inquiries  made  by  teachers  have  been  embodied  in  an  Informa- 
tion Blank  which  was  sent  to  each  school  giving  the  tests.  This 
blank  was  returned  by  all  except  schools  IV,  XII,  XIII  and  XXII. 
The  data  thus  gathered  is  included  in  this  appendix. 

I.   A  BRIEF  STATEMENT  OF  DIRECTIONS  FOR  SCORING  PAPERS1 

Test  A. — i.  The  positive  values  assigned  to  exercises  I,  II, 
III,  IV  and  V,  of  Test  A  are  15,  17,  19,  23  and  26  respectively. 
The  necessary  steps  for  a  perfect  answer  to  each  exercise  are 
given  on  pages  28-31.  The  pupil's  positive  score  is  obtained 
by  marking  each  exercise  on  the  basis  of  the  value  assigned  to  it 
and  taking  the  sum  of  such  marks. 

2.  The  negative  score  is  the  total  number  of  incorrect  and 
unnecessary  drawings  in  his  paper. 

3.  If  a  line  or  a  part  of  a  figure  ought  to  fulfill  two  or  more 
conditions  but  the  pupil  has  drawn  it  to  fulfill  only  a  part  of 
these  conditions,  credit  is  given  for  the  correct  points  and  the 
incorrect  points  are  counted  in  the  negative  score. 

4.  Any  unnecessary  drawings  are  to  be  counted  in  the  negative 
score.     In  particular,   if  in  exercise  IV,   the  pupil  draws   the 
bisectors  of  the  interior  and  exterior  vertical  angles  at  each  of 
the  three  vertices,  the  drawings  at  two  of  them  are  counted  as 
unnecessary. 

5.  The  lettering  of  figures  is  not  considered  when  scoring  the 
papers  unless  A  and  B  are  incorrectly  used  in  exercise  I. 

6.  If  a  pupil  draws  a  special  figure  for  any  one  of  the  exercises 
but  draws  it  correctly,  full  credit  is  given. 

7.  If  in  exercise  III,  the  medians  are  not  produced  to  the  mid- 
points of  the  opposite  side  but  would  pass  through  such  points 
if  produced,  full  credit  is  given  for  the  drawing. 

1  For  a  fuller  discussion  of  the  directions  for  scoring  see  pages  28-43. 

97 


98  INVESTIGATION   OF   CERTAIN   ABILITIES 

Test  B. — I.  The  positive  values  assigned  to  exercises  I,  II, 
III  and  IV  of  Test  B  are  21,  23,  26  and  30  respectively.  The 
necessary  steps  for  a  perfect  answer  to  each  exercise  are  given 
on  pages  32-34.  The  pupil's  positive  score  is  determined  by 
marking  the  hypothesis  and  conclusion  of  each  exercise  separately 
on  the  basis  of  the  value  assigned  to  it  and  then  averaging  these 
two  marks.  The  sum  of  these  averages  for  all  the  exercises  of 
the  test  is  the  pupil's  final  positive  score. 

2.  The  negative  score  is  the  total  number  of  incorrect  and 
unnecessary  statements  in  the  entire  test. 

3.  A  statement  is  counted  as  correct  only  when  it  is  given 
correctly  in  terms  of  the  figure,  but  care  must  be  taken  not  to 
count  off  twice  for  the  same  lack  of  specific  statement. 

4.  General  statements  not  given  in  terms  of  the  figure  are  not 
counted  in  determining  the  negative  score  unless  such  statements 
are  given  incorrectly. 

5.  Credit  is  not  given  for  parts  of  the  hypothesis  stated  in  the 
conclusion  excepted  as  noted  in  6  below. 

6.  If   in    exercise    IV    the    pupil    has    included    CD  =  \AB, 
CD  >  \AB,  CD  <  \AB  correctly  as  conditional  clauses  in  the 
conclusion,  full  credit  for  each  statement  as  a  part  of  the  hypo- 
thesis is  given. 

Test  C. — I.  The  positive  values  assigned  to  exercises  I,  II,  III 
and  IV  of  Test  C  are  25,  24,  27  and  24  respectively,  and  the 
numbers  of  correct  statements  considered  as  perfect  answers 
are  8,  30,  7  and  18  respectively.  The  pupil's  positive  grade  is 
obtained  by  marking  each  exercise  on  the  basis  of  the  value 
assigned  to  it  and  taking  the  sum  of  these  marks  for  all  the 
exercises  of  the  test. 

2.  The  negative  score  is  the  sum  of  all  incorrect  statements 
in  the  entire  test. 

3.  A  statement  is  counted  as  correct  only  when  it  gives  a 
geometrical  relation  or  a  magnitude  correctly  in  terms  of  the 
figure. 

4.  General  statements  not  given  in  terms  of  the  figure  are  not 
counted  in  determining  the  negative  score  unless  they  are  in- 
correctly stated. 

5.  Full  credit  is  given  for  all  facts  included  in  a  continued 
equation  or  inequality,  but  double  credit  is  not  given  for  state- 
ments repeated  in  the  same  or  slightly  different  forms. 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY  99 

6.  If  a  pupil  makes  additional  drawings  all  statements  involv- 
ing such  drawings  are  to  be  eliminated. 

Test  D. — I.  The  positive  values  assigned  to  exercises  I,  II 
and  III  of  Test  D  are  27,  34  and  39  respectively.  The  number  of 
steps  in  a  correct  answer  depends  upon  the  method  of  proof 
used  by  the  pupil.  These  steps  are  given  on  pages  38-41  for 
all  proofs  found  in  the  papers  graded  by  the  author.  The 
pupil's  positive  score  is  obtained  by  marking  each  exercise  on 
the  basis  of  the  value  assigned  to  it  and  adding  these  marks  for 
all  exercises  of  the  test. 

2.  The  negative  score  is  the  sum  of  all  incorrect  and  unneces- 
sary statements  in  the  entire  test. 

3.  Any  reasons  or  authorities  for  the  various  steps  in   the 
proof  are  to  be  disregarded. 

4.  If  a  statement  is  out  of  its  logical  order  it  is  counted  as 
incorrect  unless  its  relation  is  indicated  in  some  way  (e.  g.,  by 
"for,"  "since,"  etc.). 

5.  If  a  proof  is  incomplete,  the  pupil  should  receive  credit  for 
the  number  of  correct  steps  given.     If  it  is  possible  to  complete 
the  pupil's  proof  in  more  than  one  way,  the  smallest  possible 
number  of  steps  is  taken  as  the  required  number  for  a  perfect 
answer. 

Test  E. — i.  The  positive  values  assigned  to  exercises  I,  II, 
III  and  IV  of  Test  E  are  12,  16,  33  and  39  respectively.  If  the 
drawings  made  by  a  pupil  for  an  exercise  make  a  proof  possible, 
the  positive  score  is  the  full  value  assigned  to  that  exercise. 
Otherwise  it  is  zero.  The  sum  of  the  marks  for  all  exercises  of 
the  test  is  the  pupil's  final  positive  score. 

2.  The  negative  score  is  the  sum  of  all  incorrect  and  unneces- 
sary drawings  in  the  entire  test. 

3.  Any  drawing  which  leads  to  a  proof  is  counted  as  correct. 
Any  additional  drawings  are  considered  as  unnecessary. 

II.  INFORMATION  BLANK 

The  following  is  a  copy  of  the  information  blank  which  each 
school  was  requested  to  fill  out  and  return  to  the  author. 

INFORMATION  BLANK 

The  head  of  the  Mathematical  Department  will  please  give  the  following 
information  concerning  the  classes  tested: 


100  INVESTIGATION   OF   CERTAIN   ABILITIES 

1.  What  text  in  geometry  was  used? 

2.  How  many  weeks  were  given  to  the  first  two  books  of  geometry? 

3.  How  many  recitations  per  week  were  given  to  geometry? 

4.  In  which  year  of  the  high  school  course  were  the  first  two  books  of  geometry 
studied? 

5.  How  much  algebra  did  the  pupils  have  before  beginning  the  course  in  formal 
geometry? 

6.  Had  the  pupils  had  a  preliminary  course  in 

a.  Experimental  geometry? 

b.  Constructional  geometry? 

7.  How  long  were  these  preliminary  courses? 

8.  When  were  these  preliminary  courses  given? 

9.  State  briefly  the  method  of  instruction  used.     (Especially  any  features  of 
the  method  which  would  affect  the  results  of  the  test  given.) 


10.  On  the  reverse  side  of  this  sheet  give  any  additional  facts  which  you  think 
would  influence  the  results  of  the  tests. 

11.  Name  of  school 

12.  Name  of  person  giving  this  information 

III.  THE  TEXT  BOOK 

Each  of  the  following  text  books  was  used  by  one  or  more  of 
the  schools  giving  the  tests : 

1.  Durell  8.  Phillip  and  Fisher 

2.  Durell  and  Arnold  9.  Robbins 

3.  Ford  and  Ammerman  10.  Schultze  and  Sevenoak 

4.  Hart  and  Feld man  n.  Stone-Millis 

5.  Lyman  12.  Wentworth 

6.  Milne  13.  Wentworth  and  Smith 

7.  Palmer  and  Taylor  14.  Wells 

15.  Wells  and  Hart 

Table  XL  indicates  the  text  book  used  by  each  school  and 
also  the  test  which  was  given  in  that  school.  The  numbers  refer 
to  the  books  given  above.  Thus,  school  VI  in  which  Test  E 
was  given  used  two  books;  namely,  Durell  and  Wells.  The 
data  of  this  table  has  no  scientific  value  and  no  conclusions 
should  be  drawn  from  it.  It  only  answers  the  question  so  often 
asked  by  teachers  who  gave  the  tests,  "What  text  is  used  by  a 
given  school?" 


FUNDAMENTAL   TO   THE    STUDY   OF   GEOMETRY 


101 


TABLE  XL. — Text  book  used  by  each  school. 


Test  A 

Test  B 

TestC 

TestD 

TestE 

School 

Text 

School 

Text 

School 

Text 

School 

Text 

School 

Text 

XXIII 

13 

XIV 

13 

VII 

13 

V 

ii 

I 

4 

XXV 

I 

XV 

13 

VIII 

IO 

XXIII 

13 

II 

10 

XXXIII 

13 

XVI 

13 

IX 

13 

XXIV 

13 

III 

10 

XXXV 

13 

XVII 

13 

X 

13 

XXV 

i 

V 

II 

XXXVI 

13 

XVIII 

13 

XI 

13 

XXVI 

4 

VI 

1,  14 

XXXIX 

4 

XIX 

4 

XXVIII 

13 

XXVII 

9 

VII 

13 

XL 

13 

XX 

10 

XXXII 

7' 

XXIX 

8 

LV 

13 

XLI 

IS 

XXI 

13 

XXXIV 

13 

XL  VI 

13 

XLII 

6 

XXV 

i 

XXXVII 

13 

XLIII 

9 

XXX 

13 

XXXVIII 

3 

XLIV 

15 

XXXI 

4 

XLV 

2 

XL  VI  I 

13 

L 

13 

XL  VI  II 

2 

LI 

13 

XLIX 

2 

LI  I 

13 

LVII 

13 

LIII 

5.  12 

LVI  1  1 

13 

LIV 

I 

LVI 

13 

LIX 

4,8 

LX 

2 

LXI 

13 

LXII 

5 

LXIII 

i 

IV.  THE  AMOUNT  OF  TIME  GIVEN  TO  THE  FIRST  Two  BOOKS 

OF  GEOMETRY 

The  number  of  weeks  given  to  the  first  two  books  of  geometry 
varies  from  n  to  33.  Most  schools  devote  five  recitations  per 
week  to  geometry,  while  some  devote  only  four,  and  a  very  few 


S3 


§1 


H  -  >i  a  H  a 


FIG.  24.     Relation  of  time  devoted  to  the  first  two  books  of  geometry  to  the 
median  scores  made  on  Test  A. 

=  Number  of  recitations  devoted  to  the  first  two  books. 

o-o-o-o  =  Median  positive  score  made  by  each  school. 
=  Median  negative  score  made  by  each  school. 


102 


INVESTIGATION   OF   CERTAIN   ABILITIES 


TABLE  XLI. — Number  of  recitations  given  to  the  first  two  books  of  geometry. 


Test  A 

TcstB 

TestC 

TestD 

Test  E 

School 

No. 
Rec. 

School 

No. 
Rec. 

School 

No. 
Rec. 

School 

No. 
Rec. 

School 

No. 
Rec. 

XXIII 

90 

XIV 

80 

VII 

80 

V 

90 

I 

IOO 

XXV 

IOO 

XV 

IOO 

VIII 

IOO 

XXIII 

90 

II 

I2O 

XXXIII 

100 

XVI 

IOO 

IX 

92 

XXIV 

90 

III 

IOO 

XXXV 

90 

XVII 

IOO 

X 

80 

XXV 

IOO 

V 

90 

XXXVI 

90 

XVIII 

105 

XI 

IOO 

XXVI 

95 

VI 

IOO 

XXXIX 

120 

XIX 

70 

XXVIII 

115 

XXVII 

95 

VII 

80 

XL 

80 

XX 

105 

XXXII 

48 

XXIX 

80 

LV 

IOO 

XLI 

95 

XXI 

105 

XXXIV 

80 

XLVI 

IIO 

XLII 

105 

XXV 

IOO 

XXXVII 

90 

XLIII 

80 

XXX 

85 

XXXVIII 

105 

XLIV 

no 

XXXI 

80 

XLV 

us 

XLVII 

1  20 

L 

no 

XL  VIII 

125 

LI 

70 

XLIX 

65 

LII 

120 

LVII 

no 

LIII 

IOO 

LVI  II 

150" 

LIV 

105 

LVI 

90 

LIX 

no 

LX 

IOO 

LXI 

IOO 

LXII 

us 

LXIII 

55 

only  three  recitations  to  the  subject.  Table  XLI  indicates  the 
total  number  of  recitations  given  to  the  first  two  books  of  ge- 
ometry by  each  school.  In  the  case  of  schools  XVIII  and  XIX 
the  indefinite  way  in  which  the  data  were  given  made  it  im- 
possible to  do  more  than  give  approximate  results. 


FIG.  25.     Relation  of  time  devoted  to  the  first  two  books  of  geometry  to  the 
median  scores  made  on  Test  B. 


FUNDAMENTAL  TO  THE   STUDY  OF  GEOMETRY 


103 


The  number  of  periods  devoted  to  the  first  two  books  of 
geometry,  the  positive  scores,  and  the  negative  scores  for  Tests 
A,  B,  C,  D  and  E  are  represented  graphically  in  Figs.  24  to  28 
respectively.  If  there  were  a  positive  correlation  between  the 


FIG.  26. 


*    £   H   H   M 

Relation  of  the  time  devoted  to  the  first  two  books  of  geometry  to 
the  median  scores  made  on  Test  C. 


number  of  recitations  and  the  test  scores,  then  the  curve  for  the 
positive  scores  would  fall  and  that  for  the  negative  scores  would 
rise  as  the  curve  for  the  number  of  recitations  falls.  Figure  24 
shows  that  there  is  but  slight,  if  any,  relation  between  the  scores 


FIG.  27.     Relation  of  time  devoted  to  the  first  two  books  of  geometry  to  the 
median  scores  made  on  Test  D. 

for  Test  A  and  the  number  of  recitations.  Figure  25  shows  a 
similar  condition  to  exist  for  the  positive  scores  for  Test  B,  while 
there  is,  perhaps,  a  slight  positive  correlation  between  the 
negative  scores  and  the  number  of  recitations.  From  Fig.  26  it 
appears  that  the  larger  the  number  of  recitations  given  to  the 
first  two  books  of  geometry  the  fewer  are  the  facts  which  the 


104  INVESTIGATION   OF   CERTAIN   ABILITIES 

pupils  are  able  to  recall.  However,  the  number  of  errors  tends 
to  decrease  as  the  number  of  recitations  increases.  In  the  case 
of  Test  D  (Fig.  27)  the  number  of  recitations  seems  to  bear  a 
slight  positive  relation  to  the  positive  scores  and  a  slight  negative 
relation  to  the  negative  scores.  In  Fig.  28  it  is  difficult  to  detect 
any  relation  between  the  test  scores  and  the  number  of  recitations. 


FIG.  28.     Relation  of  time  devoted  to  the  first  two  books  of  geometry  to  the 
median  scores  made  on  Test  E. 

It  must  be  remembered  that  the  number  of  schools  tested  is  too 
small  and  the  conditions  in  these  schools  are  too  varied  to  permit 
of  any  definite  conclusions  being  drawn.  However,  these  data 
do  raise  a  question  as  to  whether  excessive  time  spent  on  the 
first  two  books  of  geometry  is  justifiable  in  so  far  as  the  abilities 
tested  are  concerned. 

V.  THE  PLACE  OF  GEOMETRY  IN  THE  CURRICULUM 

Table  XLII  indicates  the  year  in  which  the  first  two  books  of 
geometry  are  given  in  each  of  the  schools  returning  the  Informa- 
tion Blank.  For  the  purpose  of  comparison  the  median  scores 
are  also  given.  A  large  majority  of  the  schools  give  the  first 
two  books  of  geometry  some  time  during  the  second  year  of  high 
school.  In  fact  there  is  so  little  variation  from  this  that  we  can 
draw  no  conclusions  as  to  what  is  the  best  time  to  begin  the 
study  of  formal  geometry.  However,  it  is  noteworthy  that 
school  V  made  among  the  highest  scores  although  it  began  the 
study  of  geometry  during  the  first  year  of  high  school,  and  that 
no  school  which  began  the  study  in  the  third  year  made  an 
exceptionally  high  score.  This  raises  (but  does  not  answer)  a 
question  as  to  whether  it  is  best  to  put  off  the  study  of  plane 
geometry  until  the  more  advanced  years  of  the  high  school. 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY 


105 


ajoog         oo  q  xo  q\  oo  o  ro 
N  N  M  cJ  M  PO 

aaoog         v  O  ^t  o  t-  cs  ro 

9AIJISOJ  vO   O  O     ^   N 

JB3A 

siooqog 
^  a-ioos 

p  10  oo  o  o 

Sb 

^     Q 

!  * 

s[0oqog 

I" 
s 

ajoog 

a 

6<o 

SJOOg 
SAIJISOJ 

*        O   ' 

1    ^ 

-s  ^SG^X 

1 

.18 

oqoo  M  q  100  lortoo  ro>ooq 
ro  «  ro  «  ro  cj 

>sj        ^        'Apisoj;        ror>i>ioiooo  t^^-vo  rj-io\o 

C"H 

3 

Sc 

w  sl0oqoS 

5 
•< 
H 

*o  o\  ^O   M   to  oo   O\  r^*  O   ^"  oo   W5  O   fC  r**  oo  ^5  oo  O   *^t"  w  \o  O 
ooO^O   r^o   ^txA^)iot^-i>ooooO  iAw\o  V)I>M   10 

M 

WODC         9  ^^  ^9  ^P^^^^P^^P^  9  9°9  9°9  ^  ^^ 

i1^ 


EE^^XXSnHE 


106  INVESTIGATION   OF   CERTAIN   ABILITIES 

VI.  THE  AMOUNT  OF  TIME  DEVOTED  TO  ALGEBRA  BEFORE  BE- 
GINNING THE  STUDY  OF  PLANE  GEOMETRY 

In  answer  to  question  5  of  the  Information  Blank  most  schools 
stated  the  amount  of  time  devoted  to  algebra  before  the  study 
of  geometry  was  begun.  A  few  schools  stated  the  amount  of 
work  done.  In  order  to  make  the  data  comparable  we  have,  in 
the  latter  cases,  replaced  the  amount  of  work  done  by  the  time 
usually  required  to  do  that  work,  although  we  realize  that  schools 
vary  as  to  the  amount  of  time  devoted  to  a  given  amount  of 
work.  The  data  from  question  5  together  with  the  median 
scores  are  given  in  Table  XLIII.  A  study  of  this  table  shows 
that  an  increased  length  of  time  given  to  the  study  of  algebra 
does  not  necessarily  mean  an  increase  in  ability  to  do  these  tests. 
This  result  is  to  be  expected,  for  the  abilities  investigated  in  this 
study  have  but  little  or  no  relation  to  algebra  as  now  taught  in 
most  of  our  high  schools. 

VII.  EXPERIMENTAL  AND  CONSTRUCTIONAL  GEOMETRY 

Only  five  schools  reported  that  anything  in  the  way  of  experi- 
mental or  constructional  geometry  had  been  done  before  the 
pupils  began  their  study  of  formal  geometry.  The  pupils  of 
schools  III  and  LIX  had  a  half  year  of  constructional  geometry 
during  the  first  year  of  high  school,  and  school  VI  gave  three 
recitations  per  week  during  the  second  half  of  the  second  year. 
In  school  XVIII  some  constructional  work  was  given  in  con- 
nection with  the  algebra  of  the  first  year,  and  school  XLIX 
devoted  the  first  four  weeks  of  the  second  year  to  constructional 
geometry. 

VIII.  THE  METHODS  USED 

The  answers  to  question  9  were  so  varied  that  it  was  impossible 
to  classify  them  except  in  a  very  rough  way.  Twenty-one  of  the 
schools  returning  the  Information  Blank  did  not  answer  this 
question.  Twenty-eight  schools  discussed  the  class  manage- 
ment rather  than  methods,  while  the  answers  of  twenty-seven 
contained  statements  relative  to  the  content  of  the  course.  Six 
schools  indicated  their  methods  by  such  general  terms  as  syllabus, 
synthetic,  inductive,  analytic  and  heuristic.  Only  six  of  the 
schools  indicated  that  their  methods  were  directly  intended  to 


FUNDAMENTAL  TO  THE  STUDY  OF  GEOMETRY 


107 


oo  O  10  q\oo  O   ro 

N     01     N     M     N     M     CO 


3AUISOJ 


M    N    M    c>  od 

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108  INVESTIGATION    OF   ABILITIES 

develop  the  abilities  with  which  this  study  is  concerned.  In  fact, 
if  we  may  judge  from  the  answers  to  question  9,  it  is  doubtful 
if  the  methods  of  many  teachers  are  suited  to  the  development 
of  these  abilities. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  5O  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


JUL    221941 


LD  21-100m-7,f40 (6936s) 


M 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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